The relaxation method has enjoyed an intensive development during many decades and this new edition of this comprehensiv
262 49 5MB
English Pages 601 [602] Year 2020
Table of contents :
Contents
Preface
Preface to the second edition
1 Background Generalities
2 Theory of Convex Compactifications
3 Young Measures and Their Generalizations
4 Relaxation in Optimization Theory
5 Relaxation in Variational Calculus: Scalar Case
6 Relaxation in Variational Calculus: Vectorial Case
7 Relaxation in Game Theory
8 Relaxation in evolutionary problems
Bibliography
List of Symbols
Index
Tomáš Roubíček Relaxation in Optimization Theory and Variational Calculus
De Gruyter Series in Nonlinear Analysis and Applications
Editor in Chief Jürgen Appell, Würzburg, Germany Editors Catherine Bandle, Basel, Switzerland Alain Bensoussan, Richardson, Texas, USA Avner Friedman, Columbus, Ohio, USA Mikio Kato, Tokyo, Japan Wojciech Kryszewski, Torun, Poland Umberto Mosco, Worcester, Massachusetts, USA Simeon Reich, Haifa, Israel Vicenţiu D. Rădulescu, Krakow, Poland
Volume 4
Tomáš Roubíček
Relaxation in Optimization Theory and Variational Calculus 2nd Edition
Mathematics Subject Classification 2020 Primary: 4902; 49J, 49K, 54D35, 90C, 91A. Secondary: 34H05, 34H05, 35Q93, 37N40, 46A55, 46N10, 65K10, 74B20, 74N, 78A30.
Author Prof. Ing. Tomáš Roubíček, DrSc. Mathematical Institute Faculty of Mathematics & Physics Charles University Sokolovská 83, CZ186 75 Praha 8 and Institute of Thermomechanics Czech Academy of Sciences Dolejškova 5, CZ182 00 Praha 8 Czech Republic
ISBN 9783110589627 eISBN (PDF) 9783110590852 eISBN (EPUB) 9783110589740 ISSN 0941813X
Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2020 Walter de Gruyter GmbH, Berlin/Boston Typesetting: Tomáš Roubíček Printing and binding: CPI books GmbH, Leck www.degruyter.com
Ê
To the memory of Marie and Dr. Ervin Robits hek, vi tims of the Holo aust.
Contents Prefa e Ê XI Prefa e to the se ond edition Ê XVIII 1
Ê1 Ê1
Ba kground Generalities
1.1
Order and topology
1.2
Linear, nonlinear, and onvex analysis
Ê 10
Ê9
1.2.a
Linear fun tional analysis
1.2.b
Convex sets
1.2.
Means of ontinuous fun tions
1.2.d
Solving abstra t nonlinear equations
1.3
Ê 14
Fun tion and measure spa es
Ê 18 Ê 22
Ê 25 Ê 26
1.3.a
Bo hner and Lebesgue spa es
1.3.b
Spa es of measures
1.3.
Spa es of smooth fun tions and Sobolev spa es
1.4 1.4.a 1.4.b 1.4. 1.4.d 1.5 1.5.a 1.5.b 1.5. 1.5.d
2 2.1
Ê 30
Some dierential and integral equations
Ê 38
Ordinary dierential and dierentialalgebrai equations
Ê 45 Partial dierential equations of paraboli type Ê 50 Integral equations of Hammerstein type Ê 54 Basi s from optimization theory Ê 55 Existen e, stability, approximation Ê 55 Optimality onditions of the 1st order Ê 61 Multi riteria optimization Ê 70 Non ooperative game theory Ê 72 Partial dierential equations of ellipti type
Theory of Convex Compa ti ations Ê 81 Convex ompa ti ations Ê 82
Ê 84
2.2
Canoni al form of onvex ompa ti ations
2.3
Convex
2.4
Approximation of onvex ompa ti ations
2.5
Extension of mappings
2.6
Inverse systems of onvex ompa ti ations
3 3.1 3.1.a 3.1.b 3.1.
Ê 33
 ompa ti ations Ê 93
Ê 106
Young Measures and Their Generalizations Classi al Young measures Ê 118 Basi s enario and results Ê 118 Some illustrations Ê 131 Some more results Ê 133
Ê 103 Ê 111
Ê 117
Ê 38
VIII 3.2
Ë
Contents
Various generalizations
Ê 135 Ê 136
3.2.a
Generalization by Fattorini
3.2.b
Generalization by S honbek, Ball, Kinderlehrer and Pedregal
3.2. 3.2.d 3.3 3.3.a 3.3.b 3.3. 3.3.d 3.3.e 3.4 3.5 3.5.a 3.5.b 3.5. 3.5.d 3.6 3.6.a 3.6.b 3.6.
4 4.1
Ê 146 L1 spa es Ê 163 p A lass of onvex ompa ti ations of balls in L spa es Ê 166 p Generalized Young fun tionals YH % ; S Ê 166 The omposition h Ǳ Ê 174 Some on rete examples Ê 176 Coarse polynomial ompa ti ation by algebrai moments Ê 186 p Ê 188 Compatible systems of Young fun tionals on B I ; L p A lass of onvex  ompa ti ations of L spa es Ê 191 Approximation theory Ê 204 A general onstru tion Ê 205 An approximation over Ê 211 An approximation over S Ê 215 Higherorder onstru tions by quasiinterpolation Ê 219 Extensions of Nemytski mappings Ê 222 Oneargument mappings: ane extensions Ê 223 Twoargument mappings: semiane extensions Ê 226 Twoargument mappings: biane extensions Ê 236 Generalization by DiPerna and Majda Fonse a's extension of
;
(
)
(
Ê 243 Abstra t optimization problems Ê 244
)
Relaxation in Optimization Theory
Ê 263
4.2
Optimization problems on Lebesgue spa es
4.3
Optimal ontrol of nitedimensional dynami al systems
Ê 277
4.3.a
Original problem
4.3.b
Relaxation s heme, orre tness, wellposedness
4.3.
Optimality onditions
4.3.d
Ê 295 Approximation theory Ê 305
Illustrative omputational simulations: os illations
4.3.f
Illustrative omputational simulations: os illations and on entrations
4.3.g
Optimal ontrol of dierentialalgebrai systems
4.4
4.4.b 4.4. 4.4.d 4.5 4.5.a 4.5.b
Ellipti optimal ontrol problems
Ê 325
Ê 277
Ê 288
4.3.e
4.4.a
Ê 138
Ê 310
Ê 315 Ê 318
Ê 325 Ê 333 Optimal ontrol of NavierStokes' equations Ê 339 Optimal material design of some stratied media Ê 342 Paraboli optimal ontrol problems Ê 346 Innitedimensional dynami alsystem approa h Ê 348 The original problem and its relaxation
Optimality onditions in semilinear ase
An approa h through paraboli partial dierential equations
Ê 355
Contents
Optimal ontrol of NavierStokes equations
4.5. 4.6
5 5.1 5.2 5.3 5.4 5.5 5.6
6 6.1 6.2 6.3 6.4 6.5 6.6 6.7
7
Optimal ontrol of integral equations
Ê 373
Ê 382 Ê 383 Relaxation of variational problems; p ¡ 1. Ê 394 Optimality onditions for relaxed problems Ê 401 Relaxation of variational problems; p # 1. Ê 410 Convex approximations of relaxed problems Ê 416 Example: Mi rostru ture in ferromagneti materials Ê 429 Convex ompa ti ations of Sobolev spa es
Relaxation in Variational Cal ulus: Ve torial Case Ê 435 Prerequisites around quasi onvexity Ê 436 Gradient generalized Young fun tionals Ê 441 Variational problems and their relaxation Ê 453 FEMapproximation Ê 458 Further approximation: an inner ase Ê 461 Further approximation: an outer ase Ê 464 Multiwell problems: illustrative al ulations Ê 469 Relaxation in Game Theory
Ê 483
Ê 484
Abstra t gametheoreti al problems
7.2
Games on Lebesgue spa es
7.3
Example: Games with dynami al systems
7.4
Example: Ellipti games
8.1
8.2.b 8.2. 8.3
Ê 507
Ê 493
Ê 513
Evolution on abstra t onvex ompa ti ations Rateindependent evolution
Ê 515
Quasistati ratedependent evolution
8.1.b
8.2.a
Ê 490
Relaxation in evolutionary problems
8.1.a
8.2
Ê 365
Relaxation in Variational Cal ulus: S alar Case
7.1
8
Ë IX
Ê 513
Ê 521
Appli ations of relaxation in rateindependent evolution Perfe t plasti ity at small strains
Ê 524
Ê 523
Ê 526 Ê 528 Notes about measurevalued solutions to paraboli equations Ê 532 Evolution of mi rostru ture in ferromagneti materials
Evolution of mi rostru ture in shapememory materials
Bibliography Ê 539 List of Symbols Ê 571 Index Ê 577
Prefa e Has not every ... variational problem a solution, provided ... if need be that the notion of a solution shall be suitably extended? [386, p.470℄
David Hilbert This
1
senten e
had
an
immense
(18621943)
inuen e
on
analysis: it led ultimately to `weak' or generalised solutions, to my generalised urves and to S hwartz distributions, ... , and then e to hattering ontrols, to mixed strategies in Game Theory, all things that we nd essential today. [809, p.241℄
Lauren e Chisholm Young
(19052000)
Let us begin with a pie e of history. In his 20th problem, David Hilbert started a fas inating development going through the whole 20th entury, namely an eort to generalize more and more the notion of solutions to various applied problems that mathemati s met. This in ludes in parti ular the al ulus of variations, ordinary and partial dierential equations, problems from optimization theory and game theory. The generalization is always based on a natural (i.e. ontinuous2) extension of the original problems; su h ontinuous extension is often addressed as relaxation.3 The rst su
ess was a generalization of the lassi al solution to dierential equations to the weak solution4, whi h admits less smoothness than ne essary to evaluate the original dierential equation in the usual sense. This is basi ally related with the theory of distributions invented later by L. S hwartz [723℄. The weak formulation of boundary and/or initial value problems for dierential equations is nowadays a
epted so generally that the original lassi al solution is often re koned as less natural. Later it was found that in some nonlinear problems one must handle beside the loss of
1
It refers to a senten e by D. Hilbert with a similar meaning as that one ited above.
2
The ontinuity always refers to some topologies and, in fa t, may mean also lower semi ontinuity,
f. Remark 2.38.
3
A tually, the word relaxation is used also in quite very many dierent o
asions with ompletely
dierent meanings, ranging from timeevolving pro esses in various rheologi al materials or in nu lear physi s or hemistry aiming towards some stressfree states or rest states or, in a ertain parallel of de reasing some stress whatever stress may mean, to relaxation in physiotherapy or re reology or psy hology (e.g. before visiting dentist or so). Even in mathemati s itself, the word relaxation is also used as iterative te hniques for solving systems of equations or as spe i handling of onstraints in mathemati al programming, or for relaxation os illations generated by nonlinear os illators.
4
First appearan e of this philosophy is probably in the work by J. Leray [488℄, where essentially the
spa es later named as Sobolev had been introdu ed. The huge development took pla e sin e fties sin e the works by S.L. Sobolev, J.L. Lions, and many other.
https://doi.org/10.1515/9783110590852201
XII
Ë
Prefa e
smoothness also two other phenomena: os illations and on entrations of the solutions. The former phenomenon was for the rst time treated in the pioneering works by L.C. Young [805807℄, followed also by E.J. M Shane [526, 527℄. The latter one was investigated typi ally in onne tion with the famous Plateau's minimal surfa e problem by a lot of authors5 . The renaissan e of Young's idea was in onne tion with generalization of solutions to some optimal ontrol problems6 and mu h later to some nonlinear partial dierential equations and non onvex variational problems arising in ontinuum me hani s7. Just re ently, DiPerna and Majda in their pioneering work [266℄ made the rst attempt to oup with on entration and os illations ee ts simultaneously. The ommon feature of this ever going generalization is to solve more and more general problems and to ensure existen e of their solutions (in a reasonable sense) in larger lasses than the original ones where the existen e an a tually fail. Typi al property that makes it possible is ompa tness. The enlarged sets, where the solutions are sought, represent thus ertain ompa ti ations of the original sets where the problems are lassi ally formulated, and the ner the ompa ti ations, the more generalized the solutions we thus obtain. This an, in prin iple, yield eventually very ne and abstra t ompa ti ations whi h are not easy to imagine, nor to use for more detailed investigations. A natural restri tion of a generality is to require the existen e of some auxiliary algebrai stru ture whi h ould be also used for a more detailed analysis. It appears useful to require the investigated ompa ti ation to be a onvex subset of some linear topologi al spa e; then we will speak about a onvex ompa ti ation.8 Typi al usage of the onvex stru ture is for optimality onditions. The development of the resear h of optimality onditions has been also en ouraged by David Hilbert [386℄, namely in his 23th problem in onne tion with the al ulus of varia
5
See Giusti [356℄ for detailed referen es.
6
The rst ontributions appeared from sixties by Gamkrelidze [343, 344℄, GhouilaHouri [352℄, M 
Shane [528℄, Medhin [531℄, Rishel [644℄ and Warga [786790℄. Re ently many other authors dealt with these so alled relaxed ontrols, e.g. Ahmed [5, 6, 8℄, Balder [5053℄, Berlio
hi and Lasry [114℄, Carlson [170℄, Chryssoverghi [209, 210℄, Goh and Teo [358, 758℄, Halanay [373℄, Fattorini [294299℄, Papageorgiou [588, 589, 591℄, Rosenblueth [651653℄, S hwarzkopf [724℄, et . Cf. also M Shane [529℄ for a survey. Another relaxation approa h has been invented by J.E. Rubio, see Remark 4.49.
7
This was initialized by Tartar [747℄, followed by a lot of other authors, espe ially by Ball [61℄, Ball
and James [63, 64℄, Chipot and Kinderlehrer [204℄, Da orogna [241, 242℄, DiPerna [265℄, Evans [287℄, Kinderlehrer and Pedregal [424, 426℄, Murat [564℄, S honbek [721℄, et .
8
In fa t, my rst attempts in late 80tieth were to deal with problems formulated on Bana h spa e
(as is indeed a typi al ase) and then rather to extent as mu h as possible (in an as mu h ontinuous way as possible) the linearspa e algebrai stru ture on the ompa ti ation, see [656℄. Later, another attempt of su h a sort has been done by J. Perán [607℄. Su h onstru tions o
ur however very umbersome and I thus developed the theory of onvex ompa ti ations not relying on the linear stru ture of original spa es, if any.
Prefa e
Ë XIII
tions. This led to the weak formulation of the EulerLagrange equation, and later also to an appropriate optimality ondition for problems involving os illation phenomena, namely the so alled EulerWeierstrass ondition for variational problems9 and the Pontryagin maximum prin iple for optimal ontrol problems.10 The essential advantage of the onvex ompa ti ation method is that the onvex geometry is used to derive the optimality onditions by a quite onventional dierential al ulus in various problems, not only in optimal ontrol.11 During the whole 20th entury, we an also observe a parallel, intensive development of the supporting bran hes of mathemati s, in parti ular of general topology, abstra t fun tional analysis, and later also nonlinear analysis and optimization theory. The purpose of this book is to ree t these a hievements and give a fairly abstra tanalysis viewpoint to the on rete problems mentioned above. Also it an be said that the presented viewpoint represents a properly nonlinear approa h be ause it forgets (to more or less extent) the original linear stru ture (if any) and imposes a new one for the relaxed problem. I believe this ree ts genuinely the fa t that original nonlinear problems themselves violate (to more or less extent) the linear stru ture of the original spa es. Let us now go briey through the ontent of the book. After Chapter 1, whi h only summarizes very briey and mostly without proofs some more or less standard needed mathemati al ba kground, the general theory of onvex ompa ti ation is introdu ed in Chapter 2. This represents an abstra t framework of our relaxation method. Then, in Chapter 3, this general onvex
ompa ti ation theory is applied to get (  or also lo ally) ompa t onvex envelopes of Lebesgue spa es, whi h represents a basi tool for relaxation of on rete problems appearing in variational al ulus and optimization of systems des ribed by dierential equations. In parti ular ases, su h envelopes an be sequentially lo ally ompa t onvex subsets, imitating thus basi attributes of nitedimensional (Eu lidean) spa es and fa ilitating usage of onventional analyti al methods.
9
For onedimensional relaxed problems it was rst formulated by Young [806℄ and M Shane [527℄,
and generalized for spe ial twodimensional ases in [807℄.
10
First it was formulated only for ordinary ontrols for system governed by ordinary dierential
equations by Russian s hool around L.S. Pontryagin, involving also V.G. Boltyanski , R.V. Gamkrelidze, and E.F. Mish henko in [125, 127, 344, 616, 617℄, following some earlier ideas of D.W. Bushaw [158℄ and M. Hestenes [382℄. Origin of these ideas an be found, however, already in works by C. Carathéodory [168℄, F.A Valentine [775℄ and K. Weierstraÿ [796℄; f. also Pes h at al. [608611℄ for histori al surveys. The extension of maximum prin iple for relaxed ontrols is due to Gamkrelidze [345℄ and Rishel [644℄ and, in a more general form by Avakov and MagarilIl'yaev [39, 40℄, Fattorini [294, 298℄, Halanay [373℄, Kaskosz [420℄, S hwarzkopf [724℄, Warga [791℄, and many others.
11
This generalizes R.V. Gamkrelidze's proof whi h used hattering ontrols in ontrast to the origi
nal derivation of maximum prin iples whi h exploited quite te hni al, so alled needle (sometimes
alled also spike) variations invented by V.G. Boltyanski and E.J. M Shane, f. [610℄ for a histori al reminis en e. Cf. [509℄ for a ertain simpli ation of su h arguments.
XIV
Ë
Prefa e
Having the tools prepared, we will be able to treat on rete problems. In Chapters 46 they will have typi ally the abstra t stru ture
(P )
J : U,Y Ù
where
J(u ; y) (u ; y) # 0 ; B(u ; y) ¢ 0 ; u ò U; y ò Y;
Minimize . 6
subje t to > 6 F
R is a ost fun tion, Y a Bana h spa e of states, U is a set of
: Y , U Ù and B : U , Y Ù 1 are mappings forming respe tively the state equation and state onstraints, and 1 are Bana h spa es, admissible ontrols, and
the latter one being ordered. The extended (relaxed) problems will then have the stru ture: (RP)
Minimize . 6
subje t to > 6 F
J (z ; y) (z ; y) # 0 ; B (z ; y) ¢ 0 ; z ò K ; z ò Z ; y ò Y;
J : Z , Y Ù R, K is a onvex set in a lo ally onvex linear topologi al spa e Z , : Z , Y Ù and B : Z , Y Ù 1 are ontinuous mappings. The original set U is to be onsidered as densely embedded into K and J , , and B as extensions of J , , and B , respe tively.
where and
The following questions will be pursued both on an abstra t level and in parti ular
ases:12
Relation between (P ) and (RP); in parti ular a so alled orre tness of the relaxation s heme.
Relations between various relaxation s hemes (RP) for a given (P ).
Existen e and stability of solutions to (RP); wellposedness of (RP).
Firstorder optimality onditions for (RP); Pontryagin's or Weierstrass' maximum prin iples.
Impa ts of results for (RP) to the original problem (P ).
Approximation theory for the relaxed problem (RP).
Numeri al implementation of approximate relaxed problems.
Going from simpler tasks to more ompli ated ones, we begin in Chapter 4 with the optimal ontrol problems, whi h ertainly represents the simplest variant of (P ), at least if the state onstraints have a reasonable stru ture. A typi al example is an optimal ontrol problem for a nonlinear dynami al system (
t ò (0; T)
represents a
time variable):
12
As the emphasis is put to relaxation method itself, a lot of other aspe ts will remain untou hed;
this in ludes higherorder optimality onditions, problems yielding a nonsmooth relaxed problems, sensitivity analysis, et .
Ë XV
Prefa e
T . 6 Minimize X ' ( t ; y; u ) d t 6 6 6 0 6 6 6 6 d y 6 6 # f(t ; y; u) for a.a. subje t to 6 6 dt
(P1 )
( ost fun tional)
t ò (0; T);
(state equation)
y(0) # y0 ; (initial ondition) u(t) ò S(t) for a.a. t ò (0; T); ( ontrol onstraints) ( t ; y ( t )) ¢ 0 for all t ò [0 ; T ℄ ; (state onstraints) y ò W 1 q (0; T; Rn ); u ò L p (0; T; Rm );
> 6 6 6 6 6 6 6 6 6 6 6 6 F
;
' : (0; T) , Rn , Rm Ù R, f : (0; T) , Rn , Rm Ù Rn , : [0; T℄ , Rn Ù y0 ò Rn , and S(t) Rm are subje ted to ertain data quali ation, W 1 q and
where
R
k,
;
L p denote respe tively the Sobolev and the Lebesgue spa es. Quite equally, the state
equation an (and will) be a nonlinear partial dierential equation, say of an ellipti or a paraboli type, or a nonlinear integral equation. In every ase, the resulted relaxed
(z ; y) # 0 admits, for any z ò K , pre isely one solution y # ( u ) and the mapping : K Ù Y , alled an problem (RP) has the property that the extended state equation
(extended) state operator, is ontinuous and even smooth. Chapter 5 is devoted to s alarvariational al ulus problems of the type
. Minimize
(P )
> F subje t to
2
X ' ( x ; y ( x ) ; y ( x ))d x
y ò W 1; p ( ) ;
Rn is a bounded Lips hitz domain and the energy density ' : , R , R Ù R satises ertain data quali ation but ' x; r; : Rn Ù R is allowed to be
where
n
(
)
non onvex. Then the relaxed variational problem has the form (RP) with
onvex
 ompa t but non ompa t, and B # 0.
linear,
K
The ve torialvariational al ulus problems of the type
. Minimize
(P )
> F subje t to
3
X ' ( x ; y ( x ) ; y ( x ))d x
y ò W 1; p ( ; m ) ;
R
' : ,R R Ù R are handled in Chapter 6; the adje tive ve torial refers m to that y is R valued, m ¡ 1. Although at the rst sight (P3 ) has the same form as (P ), (P ) is mu h more di ult than (P ) when simultaneously n £ 2 and m £ 2 and 2 3 2 m,
with
m,n
there are several essential dieren es between Chapters 5 and 6. One of them is that either
K
is non onvex or
is nonlinear and also
B #Ö 0 in general. Contrary to (P2 )
whose understanding is fairly omplete, there are still many open essential questions as far as the relaxation of (P3 ) on erns. This is basi ally onne ted with our poor understanding of quasi onvexity and related questions. In Chapters 46, two fundamental on epts (i.e. ompa tness and onvexity) have been used rather separately the former one ensured existen e and stability of solutions while the latter one enabled us to make a more detailed analysis, e.g. to pose optimality onditions.
XVI
Ë
Prefa e
However, there are appli ations with mu h more intimate onne tions between
onvexity and ompa tness. We have in mind the non ooperative game theory, or more generally the underlying S haudertype xedpoint theory, where typi ally ompa tness and onvexity are required simultaneously to ensure mere existen e of solutions. This will be the topi of Chapter 7, though it represents rather a sample of the wide area of potential appli ations. Let me onje ture here that every abstra t problem where ompa tness and onvexity plays a ertain role an reasonably be interpreted as a relaxed problem to a ertain original problem. Another usage of ompa tness together with onvexity is for evolutionary relaxed problems where hosen onvexity serves for denition of some time derivative. On top of it, the mentioned S haudertype xed point te hnique an be used here for existen e of timeperiodi solutions. This is exposed in Chapter 8.13 It should be emphasized that the relaxation method has, beside its purely mathe
inf (P ) ¡ min(RP) must be in some situations prevented while in other situations it is wel ome.
mati al aspe ts, also an essential interpretation aspe t. For example, the fa t
The former ase is typi al for variational problems where this fa t means that optimal relaxed solution annot be attained by a minimizing sequen e for the original problem. This is related with a ne essity to hold the variational onstraint
u # y exa tly.14
The latter ase appears typi ally in state onstrained optimal ontrol problems where only the state equation is to be held exa tly while the state onstraints may be held only approximately, with a ertain toleran e. Then the gap
inf (P ) ¡ min(RP) means
that relaxed ontrols an a hieve a lower ost than the original ones, whi h is naturally wel ome.15 The reader is asked for tolerating o
asionally a bit unusual notation, reated as a ompromise by unifying the standard notation from various fairly diverse elds.16 Sometimes, the very standard notation appearing in fairly dierent o
asions was kept hopefully without any onfusion.17 Bibliographi al notes are mostly mentioned as footnotes; anyhow, be ause of the wideness of the subje t, only basi referen es are provided either from the histori al purposes or just as a sour e of other referen es for a more detailed study.
13
The evolution on onvex ompa ti ations and in parti ular periodi solution existen e theory (i.e.
the whole Chapter 8) has expanded only the 2nd edition in 2019.
14
Cf. also Remark 5.38.
15
This aspe t an be ree ted by introdu ing suitable toleran es in optimization problems; f. [658℄
for a systemati pursuit of the toleran e approa h.
16
A typi al dilemma was, e.g., that
u normally stands for the ontrol variable in optimal ontrol p denotes the polyno
theory while in the variational al ulus it denotes usually the state variable,
L p spa es while in optimal ontrol it stands for the adjoint state, et . Æ stands, beside a small positive real number, also for the Dira distribution as well as for the
mial growth in
17
E.g.,
indi ator fun tion.
Prefa e
Ë XVII
Finally, I would like to mention a plenty of olleagues with whom I had a lot of fruitful dis ussions, among them espe ially Professors K. Bhatta harya, N.D. Botkin, B. Da orogna, H.O. Fattorini, J. Haslinger, J. Malý, S. Müller, I. Netuka, J.V. Outrata, W.H. S hmidt, J. Sou£ek, and V. verák. Within writing this monograph, I also beneted from the ourses I hold during the a ademi al years 1993/94 and 1995/96 at Charles University in Prague for under and graduate students. Moreover, M. Kruºík and M. Mátlová ontributed, beside areful reading and ommenting the whole manus ript, by omputer implementation of proposed algorithms and by al ulation of the examples. It is my duty and pleasure to express my deep thanks to all of them. Last but not least, spe ial gratitude is to Professor KarlHeinz Homann, Professor Jind°i h Ne£as and Dr. Ji°í Jaru²ek, who inuen ed very essentially both my intelle tual live and professional areer and thus, dire tly or not, the theme of this book. I also warmly a knowledge the hospitality of Institut für Angewandte Mathematik und Statistik (TU Mün hen), where a great deal of the book has been a
omplished. Besides, shorter stays at ENS (Lyon), IMA (Minneapolis), EPF (Lausanne), and ErnstMoritzArndtUniversität (Greifswald) were inspiring.
Praha / Mün hen, January 1996 18
18
Tomá² Roubí£ek
A tually, in 2020, few updates and ompletions were added also into this 1996prefa e to make it
more relevant regarding this 2nd edition.
Prefa e to the se ond edition The relaxation method in optimization and variational al ulus has remained an a tive topi after the rst edition was published more than two de ades ago, as it is do umented by a dozen of new spe ialized monographs.19 This se ond, revised and substantially expanded edition ree ts in parti ular few dozens of new papers of mine (and my oauthors)20 together with hundreds of other relevant referen es, without
laiming ompleteness. The main enhan ement an be spe ied as follows: First, the ba kground generalities (Chapter 1) has been rewritten or extended at several spots. The next two hapters have been enhan ed in parti ular by inverse systems of onvex ompa ti ations (Se t. 2.6 and 3.3ef), oarse polynomial onvex ompa ti ations (Se t. 3.3.d), and by a higherorder approximation using quasiinterpolation (Se t. 3.5.d). Many hanges have been done in Chapter 4: The FilippovRoxin existen e theory has been applied both to optimal ontrol of system governed by ordinarydierential equations (for what it was originally devised) and also for other optimization problems. In addition to optimal ontrol of systems governed by ordinarydierential equations where now also some omputational illustration has been added (Se t. 4.3e,f), also optimal ontrol of dierentialalgebrai systems has been presented. An illustration of usage of the maximum prin iple is there for adaptive numeri al approximation. The ellipti and paraboli equations are now more general, not relying on potentiality, and systems of su h equations being onsidered, using also simpler arguments in the proofs. As a ertain illustration, the NavierStokes system des ribing in ompressible uid ows is taken. In the ellipti equations, optimal ontrol in oe ients (optimal material design) in ertain stratied materials has been in luded in Chapter 4, too. An in remental formula is illustrated in some situations, leading to onvexity of the relaxed problems. The state onstraints are treated also in the qualied way. Beside the onventional single riteria optimization, the multi riteria Pareto/Slater on ept is applied at some spots, allowing also an interpretation as ooperative games. Variational problems in Chapters 5 and 6 have enjoyed parti ularly vast number of new results emerged in relaxation during past de ades, and only sele ted ones are re orded in this new edition. On top of it, s alar variational problems (Chap. 5) have been illustrated also by relaxation in ferromagneti materials. A newly added Chapter 8 addresses the evolution on onvex ompa ti ations, exploiting both ompa tness and onvex stru ture governed by gradient ow involving
19
Sin e the rst edition in 1997, the relaxation theme has been published in the monographs [171, 187,
192, 268, 314, 601, 603, 605, 622, 623℄, or also [33, Chap. 4,11℄, [113, Chap. 36℄, [320, Chap. 8℄, and [643, Chap. 4,7℄.
20
Spe i ally, it on erns the arti les [89, 104, 106, 141, 179, 453, 459461, 520, 536, 543, 545, 619, 675
678, 681684, 686, 688, 689, 693702, 784℄.
https://doi.org/10.1515/9783110590852202
Prefa e to the se ond edition
Ë XIX
also a dissipation potential, as devised in an older arti le [668℄ for quadrati potentials. This is addressed both on the abstra t level and on spe i appli ations for evolution of os illations (mi rostru ture) in ferroi materials or on entrations (slip bands) in perfe tly plasti materials. Simultaneous usage of onvexity and ompa tness of the relaxed problems is illustrated, beside non ooperative game theory, now also on a feasibility/ ontrollability via xedpoint theorems and on existen e of periodi solutions in evolution problems. Moreover, the notation has been o
asionally slightly modied to be more onventional or to gain a better logi . And the (rather too mu h) abstra t topologi al on ept of nets has been suppressed in favor to the more onventional on ept of sequen es, relying on an assumption of separability of spa es of test fun tions in parti ular in Chapters 48. Also, beside the mentioned enhan ements of the 1st edition, some redu tion of the presentation has o
asionally been applied. Some dieren es likely also have o
urred be ause all the nal les from the 1st edition whi h in luded all galley proofs and language orre tions, kept ex lusively by the Publisher/typesetting ompany, were unfortunately unprofessionally annihilated in the meantime and no further language orre tions have been exe uted in the old and the new text in this se ond edition. This new, extended edition also benets from the lasses on Sele ted parts from optimization theory whi h I had during 19932010 at Charles University in Prague and ree ts also some re ent resear h21. Spe ial thanks are to Miroslav Hu²ek for advising me the old generaltopologi al onstru tions of inverse systems and threads, and to my oauthors Sören Bartels, Barbora Bene²ová, and Martin Kruºík for providing the omputational al ulations and gures added to this new edition. Parti ular thanks are to Dr. Apostolos Damialis, the former A quisitions Editor for Mathemati s in W. de Gruyter in Berlin, who initiated this 2nd edition in 2017.
Praha, July 2020
21
Tomá² Roubí£ek
In this ontext, the grant proje ts 1904956S and 1929646L of the Cze h S ien e Foundation are
a knowledged.
1 Ba kground Generalities If a fun tion
Fx is ontinuous from x
# a to x # b
in lusive, then among all the values whi h it takes, ... there is always a greatest and also a smallest ...
Bernard Bolzano
(17811848)
The theory of operators ... has penetrated several highly important areas of mathemati s in an essential way. ... The theory often makes possible altogether unforeseen interpretations of the theorems of set theory or topology.
Stefan Bana h
(18921945)
This hapter is to remind sele ted fundamental on epts and results on erning general topology, fun tional analysis, and optimization and game theory. Besides this abstra t topi s, some results from theory of fun tion spa es, theory of means on spa es (or rings) of ontinuous fun tions, and from dierential and integral equations will be summarized too. By no means this hapter is intended as an survey of these elds be ause only items needed frequently throughout the book are in luded here. Also the generality is rather restri ted to the level whi h is a tually needed in what follows. Moreover, some notions needed only lo ally have not been in luded into this
hapter, and will be reminded as footnotes at relevant pla es in the further hapters. As the reader is supposed to have a basi knowledge from general topology, fun tional analysis, and fun tion spa es, most of the results in this hapter are presented without any proofs. Some others, though being more or less also quite standard, are a
ompanied by (at least sket hed) proofs. As a result, this hapter is intended rather for a onsultation via Index within reading the further hapters but not for a thorough systemati study. Moreover, basi settheoreti al notions like relations, mappings, inverse mappings, Cartesian produ ts, et ., are supposed well known and will not be spe i ally dened here at all.
1.1
Order and topology
In this se tion we will briey summarize fundamental ideas and results on erning ordered sets and general topology.1
1
For more details the reader is referred, e.g., to the monographs by Bourbaki [144℄, e h [190℄, Csaszar
[240℄, Engelking [284℄, Köthe [436℄, and Kuratowski [471℄.
https://doi.org/10.1515/9783110590852001
2
Ë
1 Ba kground Generalities
¢, on a set X will be alled ordering if it is reexive x ¢ x for any x ò X ), transitive (i.e. x1 ¢ x2 & x2 ¢ x3 imply x1 ¢ x3 for any x1 ; x2 ; x3 ò X ) and antisymmetri (i.e. x1 ¢ x2 & x2 ¢ x1 imply x1 # x2 ). The set A binary relation, denoted by
(i.e.
equipped with the ordering will be alled ordered. The ordering
¢ is alled linear if
x1 ¢ x2 or x2 ¢ x1 always hold for any x1 ; x2 ò X . An ordered set X is alled dire ted if for any x 1 ; x 2 ò X there is x 3 ò X su h that both x 1 ¢ x 3 and x 2 ¢ x 3 . Instead of x 1 ¢ x 2 , we will also write x 2 £ x 1 . By x 1 x 2 we will understand2 that x1 ¢ x2 but x1 #Ö x2 . Having two ordered sets X1 and X2 and a mapping f : X1 Ù X2 , we say that f is nonde reasing (resp. nonin reasing) if x 1 ¢ x 2 implies f ( x 1 ) ¢ f ( x 2 ) (resp. f ( x 1 ) £ f ( x 2 )). We say that x 1 ò X is the greatest element of the ordered set X if x 2 ¢ x 1 for any x2 ò X . Similarly, x1 ò X is the least element of X if x1 ¢ x2 for any x2 ò X . We say that x 1 ò X is maximal in the ordered set X if there is no x 2 ò X su h that x1 x2 . Let us note that the greatest element, if it exists, is always maximal but not
onversely. Similarly, x 1 ò X is minimal in X if there is no x 2 ò X su h that x 1 ¡ x 2 . The ordering ¢ on X indu es also the ordering on a subset A of X , given just by the restri tion of the relation ¢. We say that x 1 ò X is an upper bound of A X if x 2 ¢ x 1 for any x 2 ò A . Analogously, x 1 ò X is alled an lower bound of A if x 1 ¢ x 2 for any x2 ò A. If every two elements x1 ; x2 ò X possesses both the least upper bound and the greatest lower bound, denoted respe tively by sup( x 1 ; x 2 ) and inf( x 1 ; x 2 ) and alled the supremum and the inmum of { x 1 ; x 2 }, then the ordered set ( X ; ¢) is alled a latti e. Then the supremum and the inmum exist for any nite subset and is determined uniquely be ause the ordering is antisymmetri . If they exist for an arbitrary subsets,
X ; ¢) is alled a omplete latti e. A subset A of a dire ted set X is alled onal if for any x 1 ò X there is x 2 ò A su h that x 1 ¢ x 2 . The following assertion, though being highly non onstru tive unless X N, plays (
a fundamental role in many further onsiderations.
Lemma 1.1 (K. Kuratowski [470℄ and M. Zorn [821℄).3 If every linearly ordered subset of X has an upper bound in X , then X has at least one maximal element.
and another set X , we say that {x } ò is a net in X if there Ù X : ÜÙ x . Having another net { x } ò in X , we say that this net
Having a dire ted set is a mapping
j : Ù su h that, for any ò , it and moreover, for any ò there is ò large enough so that j ( 1 ) £ x
is ner than the net { } ò if there is a mapping holds
2
x # x j
(
)
However, in a linear topologi al spa e order by a one with a nonempty interior, the relation
will
have a bit stronger meaning, f. p. 55.
3
This assertion is equivalent to the axiom of hoi e: for every set
X
and every olle tion {
A x }xòX ,
#Ö A x X , there is a mapping f : X Ù UxòX A x su h that f(x) ò A x for any x ò X ; f., e.g., Engelking
[284, Se t. 1.4℄.
1.1 Order and topology
Ë 3
1 £ . For example, every nonde reasing mapping j : Ù su h that j( ) is onal in produ es a ner net by putting x # x j . The reader should realize
whenever
(
)
that a ner net may have the index set of stri tly greater ardinality than the original net. Having in mind a ertain property of the nets (e.g. boundedness, onvergen e,
x
et .), we will say that this property holds eventually for a net { } ò in question if there is
0 ò su h that the net {x } ò £0 ;
has this property.
Example 1.2 (Con ept of sequen es). The set of all natural numbers N ordered by the standard ordering ¢ is a dire ted set. The nets having N (dire ted by this standard ordering) as the index set are alled sequen es. Any subsequen e of a given sequen e
an be simultaneously understood as a ner net.4 A olle tion
F
of subsets of
X will be alled a lter on X if A ; B ò F
implies
A Bò
B ò F implies A ò F , and if ò Ö F . Furthermore, a olle tion B of subsets X will be alled a lter base on X if A1 ; A2 ò B implies B A1 A2 for some B ò B and if ò Ö B . For B a lter base, the olle tion { A X ; ; B ò B : A B } is a lter on X ; F , if A
of
we will say that this lter is generated by the lter base Furthermore, we will introdu e a topology of subsets of
X su h that T
le tion of sets from
T
T
B.
X , whi h will be a olle tion X itself, and with every nite ol
of a set
ontains empty set and
it ontains also their interse tion, and also with every arbitrary
olle tion of sets from
T
also their union. The elements of
T
are alled open sets (or
T open, if we want to indi ate expli itly the topology in question), while their ompleX endowed with a topology T will be alled a topologi al spa e; sometimes we will denote it by ( X ; T ) to refer to T expli itly. Having a subset A X , T A :# {A B; B ò T } is a topology on A; we will address it as a relativized
ments are alled losed. A set
topology.
T0 of subsets of X is alled a base (resp. a prebase) of a topology T T open set is a union of elements of T0 (resp. a union of nite interse tions of elements of T0 ). A olle tion
if every
x ò N X , we say that N is a neighbourhood of x if there is an open set A x ò A N . It is easy to see that the olle tion of all neighbourhoods of a given point x , denoted by N ( x ), is lter on X ; we will alled it a neighbourhood lter of x . Besides, we dene the interior, the losure, and the boundary of a set A respe tively by Having
su h that
int(A) :# {x ò X; ;N ò N (x) : N A} ;
l(A) :# {x ò X; :N ò N (x) : N A #Ö } ; bd(A) :# l(A) \ int(A) :
4
Indeed, having a sequen e { x k }kòN and its subsequen e { x k }kòN with some N N, one an put # (N; ¢), # (N; ¢), and j : Ù the in lusion N N; note that j is nonde reasing and, sin e N
is innite,
j(N) is onal in
N
, as required.
Ë
4
Having
1 Ba kground Generalities
A B X , we say that A
is dense in
B
if l(
A) B. A topologi al
spa e is
alled separable if it ontains a ountable subset whi h is dense in it.
x
X , we say that it onverges to a point is 0 ò large enough so that x ò N
Having a net { } ò in the topologi al spa e
xòX
N of x, there £ 0 ; then we say also that x is the limit point of the net in question, and write lim ò x # x or simply x Ù x . This on ept of onvergen e is alled the MooreSmith onvergen e [547℄. Let us note that x ò l( A ) if and only if there is a net in A
onverging to x ; in this ase we also say that x is attainable by a net from A . A point x ò X is alled a luster point of the net {x } ò if, for any neighbourhood N of x and for any 0 ò , there is £ 0 su h that x ò N . Obviously, every limit point is a luster point as well, but not onversely. Nevertheless, for any luster point x of a net { x } ò there exists5 a ner net { x } onverging to x . If is ri h enough, we an even onsider ò if, for any neighbourhood
whenever
; f. Example 1.4 below.
X1 ; T1 ) and (X2 ; T2 ) and a mapping f : X1 Ù X2 , "1 (A) # {x ò we say that f is ontinuous (or, more pre isely, (T1 ; T2 ) ontinuous) if f 1 X1 ; f(x1 ) ò A} ò T1 whenever A ò T2 . Alternatively, f is ontinuous if it maps every T1 onvergent net onto a T2  onvergent one. The set of all ontinuous mappings X 1 Ù X 2 will be denoted by C ( X 1 ; X 2 ); if X 2 # R endowed with the standard topology, then we n will write briey C ( X 1 ) instead of C ( X 1 ; X 2 ). For an open domain R , C ( ) an be identied with the subspa e of C ( ) onsisting from fun tions that possess ontinuous extension to the losure :# l( ). If f is a onetoone mapping and both f and "1 are ontinuous, then f is alled a homeomorphism. Also we the inverse mapping f say that f : X 1 Ù X 2 realizes a homeomorphi al embedding of X 1 to X 2 if f is a homeomorphism between X 1 and f ( X 1 ) (equipped with the relativized topology oming from X2 ). The inje tive mapping f : X1 Ù X2 is alled ontinuous (resp. dense) embedding if f is ontinuous (resp. f ( X 1 ) is dense in X 2 ). The set of all topologies on a given set X is ordered naturally by the in lusion: Having two topologies T1 and T2 on a set X , we say6 that T1 is ner than T2 or T2 is oarser than T1 if T1 T2 (or, equivalently, if the identity on X is (T1 ; T2 ) ontinuous). The X
oarsest and the nest topologies, namely { X ; } and 2 , are alled indis rete and disHaving two topologi al spa es (
rete, respe tively.
A fun tion d : X , X Ù R is alled a metri on X if, for all x 1 ; x 2 ; x 3 ò X , d(x1 ; x2 ) £ 0, d(x1 ; x2 ) # 0 is equivalent to x1 # x2 , d(x1 ; x2 ) # d(x2 ; x1 ), and d(x1 ; x2 ) ¢ d(x1 ; x3 ) % d(x3 ; x2 ). Every metri d indu es a topology T by a base {{ x ò X ; d ( x ; x 1 ) " }; x 1 ò X ; " ¡ 0}. Conversely, a topology is alled metrizable
5
6
# , N ( x ) dire ted by the ordering ¢ , and to take, for any , # ( ; N) ò # x ò N with £ ; see, e.g., Engelking [284, Proposition 1.6.1℄ for details.
It su es to put
some
x
In the literature, the notions of stronger and weaker are sometimes used in pla e of ner and
oarser, respe tively.
1.1 Order and topology
Ë 5
if there exists a metri that indu es it. However, it should be emphasized that there exist nonmetrizable topologies.7 Topologies of a given set
X may have various important properties. One of them is T
a separation property. We say that
is a T0 topology, resp. T1 topology, (sometimes
x1 ; x2 ò X there is N1 ò N (x1 ) su h that N2 ò N (x2 ) su h that x1 ò Ö N2 . If, for any x1 ; x2 ò X , N2 ò N (x2 ) su h that N1 N2 # , then T is alled a T2 
alled Kolmogorov, resp. Fré het) if for any
x2 ò Ö N1
or (resp. and) there is
there are
N1 ò N (x1 ) and
topology, or also a Hausdor topology. Every net in a Hausdor spa e may have at most
X ; T ) is alled ompletely regular8 if, x ò X and any N ò N (x), there is a ontinuous fun tion f : X Ù R su h that f(x) # 0 and f(X \ N) # 1. Eventually, a Hausdor topology T is T4 , also alled normal, if for every losed mutually disjoint subsets M ; N X there is a ontinuous fun tion f : X Ù [0; 1℄ su h that f(N) # 0 while f(M) # 1.
one limit point. A Hausdor topologi al spa e ( for any
Example 1.3 (Topologies on R).
The standard topology on
prebase omposed from all open intervals, i.e. {(
;
R whi h is indu ed by the
a ; b); a ; b ò R {"; %}}, is Haus
dor, i.e. T2 and even normal, i.e. T4 topology. If nothing is said, always we will un
R equipped with this Hausdor topology. A olle tion a ; % ; a ò R is another topology on R whi h is T but not T , f. Remark 2.38 for its usage. An example of a T topology whi h is not T is the olle tion R F ; F nite .
derstand
{(
0
1
)
}
1
{
2
Example 1.4 (Universal index set).
}
X is a ompletely regular topologi al spa e, then there is a lter base U on X,X su h that, for any x ò X , U ( x ) # {{ x ò X; ( x ; x) ò B}; B ò U } is a base of the neighbourhood lter N (x). For many investigaIt is known9 that, if
tions in ompletely regular spa es, a universal su iently ri h index set is
# U ordered by the in lusion whi h makes it dire ted. Then, for example, every
x ò l(A)
x A.10 Moreover, if a net {x } ò has a luster point x ò X , then we an laim that there exists a ner net { x } ò (using the same index set) whi h onverges to x .11
an be attained by some net { } ò
7
An example of a nonmetrizable topology is the produ t topology on
ber of Hausdor topologi al spa es
8
Completely
regular Hausdor
Xj
AjòJ
X j on an un ountable num
having at least two elements.
spa es
are
denoted
as T
3
1 spa es 2
and
sometimes
also
alled
Tikhonov spa es in the literature.
9
In fa t, it su es to take a base of any uniformity stru ture on
X ; f., e.g., Bourbaki [144, Chap. II℄
or Engelking [284, Chap. 8℄.
# B ò # U , some x su h that (x ; x) ò B. B ò U there is some B ò su h that B £ B and (x B ; x) ò B. Then it su es to put x B # x B for any B ò U .
10
Indeed, it su es to take, for
11
Indeed, we know that, for any
6
Ë
1 Ba kground Generalities
Having a topologi al spa e ( X ; T ) and taking the T0 topology from Example 1.3, a T ; T0 ) ontinuous fun tion f : X Ù R will be alled also lower semi ontinuous (with respe t to the topology T ). A fun tion f : X Ù R is alled upper semi ontinuous if (
f : X Ù R is alled ontinuous (with respe t X ) if it is both lower and upper semi ontinuous. The reader an easily verify that this ontinuity is equivalent to the (T ; T1 ) ontinuity where T1 denotes the standard topology on R indu ed by the metri d ( a 1 ; a 2 ) # a 1 " a 2 . Besides, the lower (resp. upper) semi ontinuity of f is equivalent to liminf xÙ x f ( x ) £ f ( x ) (resp. limsup xÙ x f ( x ) ¢ f ( x )), where the limit inferior and limit superior are dened respe 
"f
is lower semi ontinuous. A fun tion
to the topology
T
on
tively by
lim inf f( x ) :# sup inf f( x ) xÙ x
N òN ( x ) xò N
and
lim sup f( x ) :# inf sup f( x ) :
xÙ x
N òN ( x ) xò N
The entral topologi al notion we will rely on is the ompa tness. A topology
T
X is alled ompa t if every over of X by open subsets ontains a nite sub over. Equally12 we an dene T ompa t if every net in X has a luster point in X . Con
on
tinuous mappings map ompa t sets onto ompa t ones. On a given set, ompa t topologies are minimal in the lass of all Hausdor topologies. Every lower (resp. upper) semi ontinuous fun tion
X Ù
R on a ompa t topologi al spa e
(
X ; T ) attains
its minimum (resp. maximum), whi h is known as (a generalization of) the Bolzano
Weierstrass theorem.13 A topologi al spa e ( X ; T ) is alled sequentially ompa t if every sequen e in
X admits a subsequen e that onverges in X . A metrizable topology is
ompa t if and only if it is sequentially ompa t,14 while for nonmetrizable topologies these notions are not omparable.15 A subset
A of a topologi al spa e (X ; T ) is alled A is (sequentially) ompa t in X . A
relatively (sequentially) ompa t if the losure of topologi al spa e is alled
 ompa t if it is a union of a ountable number of ompa t
subsets, and it is alled lo ally (sequentially) ompa t if every point of its possesses a (sequentially) ompa t neighbourhood.
X j ; Tj )}jòJ of topologi al spa es, we dene the topology T on the produ t X # A j ò J X j anoni ally as the oarsest topology on X that makes (T ; T j ) ontinuous all the proje tion X Ù X j ; this topology has16 a base Having an arbitrary olle tion {(
12
See, e.g., Engelking [284, Thm. 3.1.23℄.
13
More pre isely, B. Bolzano [130℄ showed that a real ontinuous fun tion of a bounded losed in
terval is bounded. A tually, that time, the trans edental numbers were not dis overed so the lo al
ompa tness of the reals was only intuitively understood, and the on ept of what later was alled Cau hy sequen es had to be invented.
14
We refer, e.g., to Engelking [284, Thm. 4.1.17℄.
15
For examples of a non ompa t sequentially ompa t and a ompa t but not sequentially ompa t
spa es we refer to Köthe [436, Se t. 3.4℄.
16
See, e.g., Bourbaki [144, Se t. I.8.1℄ or Engelking [284, Proposition 2.3.1℄.
1.1 Order and topology
{A
jòJ
A j ; :j ò J : A j ò Tj ; & A j # X j
for all but a nite number of indi es
Ë 7
j ò J}. The fol
lowing assertion, based on the KuratowskiZorn lemma 1.1,17 is of a vital importan e:
Theorem 1.5 (A.N. Tikhonov).18 The produ t spa e (X ; T ) is ompa t if and only if all (
X j ; Tj ) are ompa t.
X ; T ), we say that a pair ( X ; i) is a ompa ti ation X into X and if X is ompa t. If, in addition, the embedding i is homeomorphi al, then X will be alled T Having a topologi al spa e (
of
X
if
i : X Ù X
is a ontinuous19 dense embedding of
onsistent. As homeomorphi al topologi al spa es are equivalent to ea h other from
X and i(X) in the T  onsistent ase. The dieren e X \ X will be addressed as a remainder. If T is a ompletely regular topology, then ( X ; T ) admits a T  onsistent ompa ti athe generaltopology viewpoint, we will sometimes not distinguish between
tion.20 In general, a ( ompletely regular) topologi al spa e may admit a large amount of (T  onsistent) ompa ti ations, so it is worth introdu ing a natural ordering of them. Having two ompa ti ations (
1
X ; i1 ) and (
2
X ; i2 ) of X , we say that the for
mer one is a ner ompa ti ation than the latter one (or, equivalently, the latter one is oarser than the former one) and write ( if there is a ontinuous mapping
:
1
XÙ
1
X ; i1 ) ³ ( 2 X ; i2 ) (or briey 1 X ³ 2 X ) i1 # i2 . 2 X xing X in the sense that
This mapping is inevitably surje tive, and we will refer to it as a anoni al surje tion.
X³
1
2
X and
X²
X , then these ompa ti ations will be alled equiv1X Ê 2 X . If 1X ³ 2 X but 1 X ÊÖ 2 X, then we will write 1 X ± 2 X , saying that 1 X is stri tly ner than 2 X (or 2 X is : 1X Ù 2X stri tly oarser than 1 X ). If 1 X ± 2 X , then the anoni al surje tion inevitably glues at least two points of the remainder of 1 X together, i.e. there are x1 ; x2 ò 1 X \ i1 (X) su h that (x1 ) # (x2 ). If a ompa ti ation is T  onsistent, then
If both
1
2
alent to ea h other and then we will write
any ner ompa ti ation is
T  onsistent, too.
Any olle tion of ompa ti ations {( j
X ; i j )}jòJ
of a given ompletely regular
spa e X admits its supremum ( X ; i ), whi h an be onstru ted21 by putting X #
l(i(X)) where i : X Ù AjòJ j X : x ÜÙ i j (x) jòJ and AjòJ j X is endowed with the standard produ t topology whi h makes it ompa t by the Tikhonov theorem. In parti ular, there exists the supremum of all ompa ti ations of
17
Noteworthy, if
J
is ountable and all (
X j ; Tj )
lemma is rather trivial so that the ompa tness of
18
X , denoted by X . This is simul
are metrizable, the usage of the KuratowskiZorn
AjòJ
Xj
has a onstru tive hara ter.
A.N. Ty hono (19061993) formulated this theorem in [772℄. Beside German spelling in this origi
nal arti le, often it is referred with dierent spelling, transliterating his name in Cyrilli , as Ty honov, Tikhonov, Tihonov, or Ti honov. See also, e.g., Bourbaki [144, Setion I.10.5℄, Dunford and S hwartz [275, Thm. I.8.5℄, Engelking [284, Thm. 3.2.4℄, Köthe [436, Se t. 3.3℄.
19
Mostly, a narrower on ept of ompa ti ations, requiring the embedding to be homeomorphi al,
is adopted in general topology; see, e.g., [190, 240, 284, 471℄. For our purposes it appears useful to a
ept su h wider on ept, the narrower on ept being spe ied by the adje tive onsistent.
20 21
See, e.g., Engelking [284, Thm. 3.5.1℄. We refer to Engelking [284, Thm. 3.5.9℄ for more details.
8
Ë
1 Ba kground Generalities
taneously the nest ompa ti ation of
tion [189, 736℄. It is always
X , being alled the e hStone ompa ti a
T  onsistent. Sometimes, (X ; T
) admits also the oarsest
T  onsistent ompa ti ation, denoted by X , and alled the Alexandro ompa ti ation. A non ompa t ompletely regular spa e admits the Alexandro ompa ti a
X is either void (if X itself X is only lo ally ompa t but non ompa t).
tion if and only if it is lo ally ompa t.22 The remainder of is ompa t) or a singleton (if
Example 1.6 (Compa ti ations of R). point ompa ti ation
If X # R, the real line, the Alexandro oneR just adds to R, gluing thus both free ends of R to
R is then homeomorphi with a ir le. Then the standard twopoint
ompa ti ation R # [" ; %℄ is stri tly ner. The nest, e hStone ompa ti ation R is still stri tly ner23 than R. Thus we get the situation gether so that
R ° Let us still remark that
R ° R :
R are here metrizable, while R is not.
R and
Another useful on ept generalizes the usual (= singlevalued) mapping: having
X2 , a mapping S : X1 Ù 2X2 , with 2X2 denoting the set of all subsets of X 2 , will be also alled a multivalued mapping from X 1 to X 2 , denoted by S : X 1 Â± X2 . Having (X1 ; T1 ) and (X2 ; T2 ) two topologi al spa es, it is worth generalizing the two sets
X1
and
on ept of ontinuity. We dene
Limsup S( x ) :# x ò X ; x1 Ù x 1 ; a net (x ; x ) Ù (x ; x 1
2
2
1
Liminf S( x ) :# x ò X ; x1 Ù x 1 : net x Ù x
1
2
1
2
2)
in
T1 , T2
: x ò S(x )Ǳ ;
(1.1a)
: x Ù x
(1.1b)
2
1
2
1
1
in
T1 ; a net x2 ò S(x1 )
2
2
in
T2 Ǳ;
The above introdu ed upper and lower limits are alled the Kuratowski limits.24
S is alled upper (resp. lower) semi ontinuous at x1 if S( x 1 ) S(x1 ) (resp. Liminfx1 Ùx1 S( x 1 ) S(x1 )). Of ourse, S is alled upsemi ontinuous25 if it is upper (lower) semi ontinuous at every x 1 ò X 1 .
The multivalued mapping Limsup x1 Ù x 1 per (lower)
22
See, e.g., Engelking [284, Theorems 3.5.1112℄.
23
In fa t, the remainder of
R
is very large, ontaining at least
2
N
2
points; f. Engelking [284,
Thm. 3.6.11℄.
24
X # %}, being denoted by the symbols Ls and Li. Let us also mention that sometimes the sym
They were invented in Kuratowski [471, Se tions 29.I and 29.III℄ for the ase of sequen es, i.e.
N
1
{
bols Limsup and Liminf may have another meaning, being dened without referring to any topology on
X2 , namely LimsupnÙ S(x n ) # V n#0 Uk#0 S ( x n%k ) and LiminfnÙ S ( x n ) # Un#0 Vk#0 S ( x n%k ),
f. [471, Se t. I.V℄.
S is alled upper (lower) semi ontinux1 ò X1 ; S(x1 ) A} is T1 open (T1  losed) for any A whi h is T2 open (T2  losed). However, both denitions oin ide with ea h other provided X 2 is ompa t, see Kuratowski [471, Se t. 43.II℄ 25
Sometimes, these notions are dened by other ways: namely
ous if the set {
or Deimling [256, Se tions 1 and 2℄.
Ë 9
1.2 Linear, nonlinear, and onvex analysis
If Limsup x1 Ù x 1 Lim x1 Ù x 1
S( x 1 ) #
S( x 1 ).
Liminf x1 Ù x 1
S( x 1 ),
we will denote this ommon limit set as
S is singlevalued (i.e. S(x) # {f(x)} for a mapping f : X1 Ù X2 ) well as the lower) semi ontinuity of S is equivalent to the usual
Let us note that if then the upper (as
ontinuity of Another
f. elegant
onstru tion
has
S. Lefs hetz:26 onsidering a dire ted set spa es
X
been
,
invented
by
a olle tion (
P. Alexandro
X ) ò
is alled an inverse system if
: £ ; 21 : X 2 Ù X 1 ontinuous : : # : 21 # identity and 2
1
1
31
: ¢ ¢ : 1
2
The mappings
3
21
inverse system as for any
and
of topologi al
#
21
32
(1.2a)
2
:
(1.2b)
from (1.2) are alled bonding mappings. Shortly, we write this
S # (X ; ) ; ò ; £ . Ea h x ò A ò X
£ . The set
is alled a thread if
S :# x ò I X ; x is a threadǱ lim ØÚ
x # x (1.3)
ò
proje tions
S # (X ; ) ; ò ; £ . We further dene the of the limit of the inverse system Pr : lim S Ù X as the restri tion on
is alled the limit of the inverse system
ØÚ
lim S of the proje tions A ò X Ù X . For all ¢ , it holds Pr # Pr . ØÚ
X in (X ; ) ò £ are Hausdor topologi al spa es, then lim S is losed ØÚ in A ò X equipped with the anoni al produ t topology; f. [284, Prop. 2.5.1℄. If all X are ompa t, this topology is ompa t by the Tikhonov theorem, so that we then have immediately the ompa tness of the inverse limit S. If all
1.2
;
;
Linear, nonlinear, and onvex analysis
In this se tion we will briey summarize fundamental ideas and results on erning linear topologi al spa es, their duals, and onvex subsets, as well as linear or nonlinear mappings or onvex fun tionals on them.27 Throughout the whole book we will
onne ourselves to topologi al ve tor spa es over the eld of reals
26
R.
The on ept of limits of the inverse systems has been invented in early 30ties of the last entury
in [486℄ exploiting a bit modied denition in [12℄ and then developed in [282℄; see e.g. [284, Se t.2.5℄ for a omprehensive exposition.
27
For more details the reader is referred, e.g., to the monographs by Choquet [208℄, Day [252℄, Dunford
and S hwartz [275℄, Edwards [278℄, Holmes [392℄, Kolmogorov and Fomin [434℄, Köthe [436℄, Taylor [751℄, Valentine [776℄ and Yosida [804℄.
10
Ë
1.2.a
1 Ba kground Generalities
Linear fun tional analysis
X ; T ) is alled a (real) linear topologi al spa e if it is equipped x1 ; x2 ) ÜÙ x1 % x2 : X , X Ù X whi h makes it a ommutative topologi al group28 and with a jointly ontinuous multipli ation by s alars ( a ; x ) ÜÙ ax : R , X Ù X satisfying (a1 %a2 )x # a1 x % a2 x, a(x1 %x2 ) # ax1 % ax2 , (a1 a2 )x # a1 (a2 x), and 1x # x. The point 0 ò X is also alled the origin. A subset K X is alled
onvex if ax 1 % (1" a ) x 2 ò K whenever x 1 ; x 2 ò K and 0 ¢ a ¢ 1, and it is alled a
one (with the vertex at the origin) if ax ò K whenever x ò K and a £ 0. A topologi al linear spa e X is alled ordered by a relation £ if this relation is an ordering and, in
A topologi al spa e (
with the binary operation (
addition, it is ompatible with the linear and topologi al stru ture in the sense that
ax £ 0 if x £ 0 and a £ 0, that x1 % x2 £ 0 if both x1 £ 0 and x2 £ 0, that x1 £ x2 implies x 1 % x 3 £ x 2 % x 3 for any x 3 , and that x £ 0 and x Ù x implies x £ 0. It is easy to see that D # { x ò X ; x £ 0} is a losed onvex one whi h does not ontain a line. Conversely, having a losed onvex one D X whi h does not ontain a line, the relation £ dened by x 1 £ x 2 provided x 1 " x 2 ò D makes X an ordered linear topologi al spa e. For a subset
A X , we dene a so alled indi ator fun tion Æ A (x) : X Ù {0; %}
by
0 %
Æ A (x) :# ® Let us note that
for
x ò A;
otherwise
:
A is onvex (resp. losed) if and only if Æ A is onvex (resp. lower semi
ontinuous).
X1 and X2 and a mapping A : X1 Ù X2 , A is a ontinuous linear operator if it is ontinuous and satises A(a1 x1 % a2 x2 ) # a1 A(x1 ) % a2 A(x2 ) for any a1 ; a2 ò R and x1 ; x2 ò X . Often we will write briey Ax instead of A(x). If X1 # X2 , a linear ontinuous operator A : X1 Ù X2 is alled a proje tor if A A # A . The set of all linear ontinuous operators X 1 Ù X 2 will be denoted by L( X 1 ; X 2 ), being itself a linear spa e when equipped with the addition and multipli ation by s alars dened respe tively by ( A 1 % A 2 ) x # A 1 x % A 2 x and ( aA ) x # a ( Ax ). As R is itself a linear topologi al spa e, we an onsider the linear spa e L( X ; R), being also denoted by X and alled the dual spa e to X . Having two topologi al linear spa es
we say that
*
The topology of a topologi al linear spa e is fully determined by a base lter of neighbourhoods of the origin
N
0}
forms a base of the topology of
N
N
0
of the
0) be ause the olle tion {x % A; x ò X ; A ò
(
X . An important lass of topologi al linear spa es
onsists of lo ally onvex spa es, having a base of
N
0) omposed from onvex sets.
(
x ÜÙ x : X Ù R is alled a seminorm if x £ 0, ax # ax, and x1 % x2 ¢ x 1 % x 2 . A fun tion x ÜÙ x : X Ù R is alled a norm if it is a seminorm and if x # 0
A fun tion
28 (
It means that this mapping is jointly ontinuous and satises
x1 % x2 ) % x3 , ;0 ò X : x % 0 # x, and :x1 ò X ;x2 : x1 % x2
# 0.
x1 % x2
# x % x , x % (x % x ) # 2
1
1
2
3
1.2 Linear, nonlinear, and onvex analysis
implies
Ë 11
x # 0. Having a olle tion of seminorms {  } ò on a linear spa e X , we an :# {{x ò X; max ò x ¢ "}; " ¡ 0; nite} is a lter base
see that the olle tion B
0
and, taking the lter generated by it as the neighbourhood lter of the origin N ( ), we obtain a lo ally onvex spa e. Conversely, every lo ally onvex spa e an be obtained by this manner if taking the olle tion of seminorms appropriately. A lo ally onvex spa e equipped with a norm and with the topology generated by this norm is alled a
normed linear spa e, its topology being also addressed as strong. Nets onverging in strong topology are alled strongly onvergent. If the olle tion of seminorms {  } ò generating the topology of a Hausdor lo ally onvex spa e is ountable (i.e. we may suppose
:# N and write {  k }kòN ) then d(x1 ; x2 ) :#
"k
H2 k #1
x " x k 1 % x "x k 1
2
1
(1.4)
2
denes a translationinvariant29 metri whi h indu es the topology of Having two normed linear spa es
L(X1 ; X2 ) by
A L X 1 (
;
X2 )
X1
and
X2 ,
X.
we an introdu e a norm on
:# sup Ax X2 ;
(1.5)
x X1 ¢1
X1 ; X2 ) a normed linear spa e. An operator A ò L(X1 ; X2 ) is alled omX1 onto relatively
ompa t subsets of X 2 . A net { x } ò in a topologi al linear spa e is alled Cau hy30 if, for any N ò N (0), there is N ò su h that x 1 " x 2 ò N for any 1 £ N and 2 £ N . If every Cau hy net
onverges in X , then X is alled omplete. A Hausdor omplete lo ally onvex spa e
whi h makes L(
pa t if it maps bounded (with respe t to the norm  X 1 ) subsets of
0
with N ( ) having a ountable base is alled a Fré het spa e, while a omplete normed linear spa e is alled a Bana h spa e. Having a lo ally onvex spa e
X and its dual X
*
, we an see that {  x * } x * ò X * with
x ; x> is a olle tion of seminorms; the bilinear form : X , X Ù R dened by < x ; x > :# x ( x ) is alled the anoni al bilinear pairing. The topology generated on X by this olle tion of seminorms is alled the weak topology and it an be seen that this topology is always oarser than the original topology and X equipped with
xx :# *
for some x ò X ; in other words, the dual *
*
*
spa e to ( X If
X
*
; weak*) is again X .
is a normed linear spa e equipped with the norm  , then
sup x ¢ is a norm on X *
*
*
1
, whi h makes
norm as the dual norm. Considering
X
its dual, denoted by
:# (X
**
*
X
*
X
*
x
ÜÙ
*
the Bana h spa e. We will refer to this
normed by the dual norm, we an think about
*
) , equipped again with the dual norm. This spa e is
X . The mapping i : X Ù X dened by :# is alled the anoni al embedding of X into its bidual, and it realizes a (weak,weak*)**
alled the bidual spa e to
*
*
as well as (strong,strong)homeomorphi al embedding.33 Moreover, the Goldstine the
B% X
orem34 says that, for dense in the ball in
ive if i ( X )
#X
**
the ball of the radius
**
% ¡ 0 in X , the image i(B % ) is weakly* X is alled reex
of the same radius. A normed linear spa e
. The reader ertainly noti ed that we have dened three lo ally on
vex topologies on
X
*
, namely the strong, the weak, and the weak* topologies. The
weak topology is always oarser35 than the strong topology, and the weak* topology is
oarser36 than the weak one. If
X
is a Bana h spa e, the Bana hSteinhaus prin iple [73℄ (often also alled
uniformboundedness prin iple, or the resonan e theorem) an be applied to ing37 that a olle tion { in
x ; ò } is bounded in X *
*
*
, say
x ; x>; ò } is bounded sequen e in X must be
provided {
£ 0 for any f £ 0, f ò F .
Proposition 1.16 (Averaging positive fun tionals
*
, i.e.
£ 0 means just
means).53 Let
F
that
be a linear sub
C0 (U) ontaining onstants. Then: The set M(F ) of all means on F an be alternatively expressed as:
spa e of (i)
Moreover, if
: F Ù R linear; £ 0 & ; 1 # 1 #:
F) :
M(
(1.22)
inf f ¡ 0, then even (f) ¡ 0 for any ò M(F ). e : U Ù M(F ) is weakly* ontinuous.
(ii) The evaluation mapping (iii)
M(F ) is weakly* ompa t and onvex subset of F * .
(iv)
M(F ) is the weak* losure of the set of all nite means.
51
For more details, the reader is referred to the monographs by, e.g., Berglund et al. [108℄, e h [190℄,
Edwards [278℄, Engelking [284℄, Gilmann and Jerison [355℄, and Yosida [804℄. The means an be dened even a bit more generally on a linear subspa e F of bounded fun tions U not ne essarily ontinuous and not ne essarily ontaining onstants. Namely, a mean is by the denition a linear fun tional F Ù su h that inf uòU f ( u ) ¢ ( f ) ¢ supuòU f ( u ); f. Edwards [278, 0 Se t. 3.5℄. This oin ides with our denition provided F C ( U ) and 1 ò F .
52
on
53
R
We refer to Berglund, Junghenn, Milnes [108, Se t. I.3℄; however, the presented assertion here is a
bit modied, e.g.
C0 (U) is not a omplex but a real algebra and F
need not be losed.
Ë
20
1 Ba kground Generalities
ò M(F ) and f £ 0, f ò F . Put fmax :# sup f(U) and fmin :# inf f(U). Obvifmin £ 0. Sin e ò M(F ), it holds
Proof. Let ously,
!! !! ; f " !!
!! 1 ( f max % f min ) ! !! 2 !
#
!! !! ¼ ; f !!
!
¢
" " "f F " " "
" 21 (fmax % fmin )½ !!!! !
*
"
" 1 (f % fmin )"""""C0 2 max (U)
1 (f "f ): 2 max min
#
0 ¢ fmin ¢ ¢ fmax . This proves £ 0 and even (f) ¡ 0 if inf f ¡ 0. : F Ù R linear su h that £ 0 and # 1. Furthermore, take f ò F and put f # " f % f C 0 U . Obviously, f £ 0 and therefore < ; f > # "< ; f > % f C 0 U £ 0. This yields < ; f > ¢ f C 0 U for any f ò F . Therefore is ontinuous, i.e. ò F , and even F # 1. Thus the point (i) has been proved. The weak* ontinuity of e : U Ù F means pre isely that u ÜÙ < e ( u ) ; f > # f ( u ) : U Ù R is ontinuous for any f ò F , whi h follows dire tly from the ontinuity of ea h f ò F C (U). This shows (ii).
Therefore,
Conversely, let us take
(
(
)
)
(
*
)
*
*
0
M(F ) is onvex and losed. By (1.20), M(F ) is ontained in the F * , and therefore, by AlaogluBourbaki theorem 1.8, it must be weakly*
In view of (1.22), unit ball of
ompa t, as laimed in the point (iii).
ò M(F ) and put M(F ) :# w* l( o(e(U))) the weak*
losure of nite means. If were not belong to M(F ), then by the HahnBana h theorem 1.11 there would exist f ò F su h that < ; f > ¡ sup r òM F < ; f >; realize also Let us go on to (iv). Take
(
)
F*
that, by Theorem 1.7, every weakly* ontinuous linear fun tional on
has the form
f ò F . However, sup òM r F < ; f > £ sup u ò U < e ( u ) ; f > # f C 0 and we obtain a ontradi tion < ; f > ¡ f C 0 U . This shows that ò M(F ). ÜÙ
for some
(
(
)
)
(
U) ,
Remark 1.17 (Conne tion between means and probability measures). From (1.22), one an easily see that in the spe ial ase F # C ( U ), the set of all means is pre isely 0
the set of all probability measures on
U.
theorem 1.32 below, M(F )
(resp. M(F )
U
Ê r a% (U) 1
is ompa t (resp. normal). If
F
Thus, in view of the Riesz representation
is smaller than
Ê rba% (U)) 1
C (U), 0
if
F
# C (U) 0
and
the means an alterna
tively be understood as lasses of probability measures with respe t to a suitable equivalen e.54 Let us now turn our attention to multipli ative means on rings of ontinuous
C (U) is alled a ring if f ; f ò R implies f f ò R , where f f denotes the pointwise multipli ation dened naturally by [ f f ℄( u ) :# f (u)f (u). Obviously, f f # f f , and thus we are talking about ommutative rings. bounded fun tions. A subspa e
R
0
1
2
1 2
1 2
1
2
1 2
1 2
2 1
C0 (U), ontains onstants and separates points from losed sets in the sense that, for every A U losed and u ò U \ A , R
ontains f being equal 1 on A and vanishing at u . As the onstant 1 represents a unit
A ring
54 if
R
is alled omplete if it is losed in
Namely, for
P U
f d1
# PU
1 ; 2 ò r a%1 (U) (resp. rba%1 (U)) we have in mind the equivalen e: 1 f d2 for any f ò F .
È
2 if and only
1.2 Linear, nonlinear, and onvex analysis
in the sense that
Ë 21
1f # f1 # f , su h omplete subrings are simultaneously so alled
ommutative unital algebras55 . Also, if (
U; T ) is ompletely regular, then C0 (U) itself
is a omplete ring. The aim is to onstru t for every ompa ti ation of of multipli ative means on a suitable ring.56 If
R
U its representation in terms C0 (U), let us denote
is a subring of
the set of averaging positive multipli ative fun tionals by
R ) :# ò M(R );
:f ; f ò R : ; f f
Mmult (
1
2
1 2
# ; f
1
; f 2 :
The elements of Mmult (R ) are also alled multipli ative means, being pre isely the
R whi h dier R ) will be endowed with the (relativized) weak* topology.
ontinuous, linear, multipli ative fun tionals on
from zero. Again,
Mmult (
Proposition 1.18 (Multipli ative means).
Let
R
be a subring of
C0 (U)
ontaining on
stants. Then: (i)
R ) is the weak* losure of e(U), and the pair (Mmult (R ); e) is a ompa ti ation of ( U; T ). Moreover, Mmult (R ) Ê Mmult ( R) with R :# lC 0 ( U ) R . If R is a omplete ring, then the ompa ti ation (Mmult (R ) ; e ) is T  onsistent. Mmult (
(ii)
(iii) Identifying equivalent ompa ti ations, there is a onetoone orderpreserving or
²
responden e between (T  onsistent) ompa ti ations of ( U; T ) ordered by and ( omplete) losed rings of ontinuous bounded fun tions ontaining onstants
ordered by the in lusion .
M(R ),
Sket h of the proof. Clearly, Mmult (R ) is weakly* losed subset of Proposition 1.16(iii) ompa t. By the Gelfand representation,57 morphi with
f () :#
. Then obviously f (e(u)) e(U) must be identi ally
is isometri ally iso
f ÜÙ f with f ò C(Mmult (R )) dened by # f(u), so that any fun tion in C(Mmult (R )) zero. Then, by Urysohn's lemma, e ( U ) must
via the mapping
vanishing on
R
hen e by
be dense in Mmult (R ).
Q : R Ù R the in lusion so that the adjoint operator Q , realizing the restri tion of linear ontinuous fun tionals from R to R , maps R onto R and is (weak*,weak*) ontinuous. Sin e two dierent fun tionals on R remain dierent after restri tion on a dense subspa e R , Q is also inje tive. Yet ontinuous oneto*
Let us denote by
*
*
*
one mapping between ompa t sets must be a homeomorphism. The point (i) has been thus proved.
55
Let us note that a subring of
C0 (U) ontaining onstants is simultaneously an algebra under the
multipli ation by s alars.
56
Let us only remark that there are several other equivalent onstru tions: the set of all maximal
ideals on su h a ring in
C0 (U), see Gelfand and at al. [348℄, or the set of all lters on U with a ertain
spe ial properties, e.g. the set of all maximal round (or alternatively ompressed round) lters with respe t to a given proximity stru ture, see e.g. Csaszar [240℄ for denitions and other details.
57
See Gelfand at al. [348℄ (where maximal ideals are used in pla e of multipli ative linear fun tionals)
or also, e.g., Yosida [804, Se t. XI.1℄.
22
Ë
1 Ba kground Generalities
lim ò e(u ) # e(u) implies T lim ò u # u T neighbourhood N of u and f ò R su h that f ( u ) # 0 and f ( U \ N ) # 1. Then, for every ò large enough one has < e ( u ) " e(u); f> 1, whi h means that u ò N be ause obviously # . and < J k u ; J k As A # A 1 % A 2 is oer ive, for % su iently large we have *
In other words, we seek
*
*
*
*
*
*
u V k # % âá
A ( u )
Suppose, for a moment, that
*
" f; u £ A(u); u " f u ¡ 0:
I k A ( u ) #Ö I k f *
*
mapping
u ÜÙ %
(1.30)
*
"1
for any
J k I k f " * " " "I k (f " *
u ò Vk
u
with V k
¢ %. Then the
" A(u)
A(u))""""V
(1.31)
*
k
u ò V k ; u ¢ %} into itself be ause J "k 1 # 1; note that V k # f V k . Also, the mapping u ÜÙ < A ( u ) ; v > : V k Ù R is ontinuous for k any v so that also u ÜÙ I A ( u ) : V k Ù V is ontinuous. By the Brouwer xedpoint k k Theorem 1.19, the mapping (1.31) has a xed point u , this means maps the onvex ompa t set {
"1 f J
*
*
*
u#%
J "k 1 I k f " A(u) : " " " " " " I k ( f " A ( u ))" "V *
(1.32)
*
*
k
As
J "k 1 f V k # f V k , (1.32) implies u V k # %. Testing (1.32) by J k u I k (f " A(u)) V k , one
gets
*
*
%2 """"I k (f " A(u))""""V # J k u ; u """" I k (f " A(u))""""V *
*
*
*
k
*
k
# % J k u ; J "k I k (f " A(u)) # % I k (f " A(u)); u # % f " A ( u ) ; I k u # % f " A ( u ) ; u 1
*
*
(1.33)
A(u) " f; u> # "% I k (A(u) " f) V k ¢ 0, a ontradi tion with (1.30). Moreover, putting v :# u k into (1.29), we an estimate62 u k u k ¢ A ( u k ) ; u k # < f; u k > ¢ f u k with a suitable in reasing fun tion : R% Ù R% su h that lim Ù () # % whi h exists due to the assumed the oer ivity (1.27) of *
whi h yields
# < f " f; u 1 " u 2 > # 0, a ontradi tion. Thus,
this limit, the identity (1.29) holds even for any
1.3
Fun tion and measure spa es
For the brevity of this se tion, we must onne ourselves only to a brief summary of basi denitions and results.63
will be a measurable subset of Rn endowed with a Lebesgue measure and 1 ¢ p ¢ %; by A we will denote the Lebesgue measure64 of a measurable subset A . We will use the standard notation for the onjugate If not said otherwise,
exponent
p/(p"1) . 6 p :# > %
if
6 F
if
1
if
1 p %; p # 1; p # % :
(1.35)
S will be a separable Bana h spa e; often S will be nitedimensional.
, we say that a property holds almost everywhere on
(in abbreviation a.e. on ) if this property holds everywhere on with the possible Besides,
Having a measurable set
63
More details an be found in the monographs by Adams [4℄, Dunford, S hwartz [275℄, Gajewski,
Gröger, Za harias [342℄, Halmos [374℄ or Kufner, Fu£ík, John [467℄. For Lebesgue spa es, see also Bourbaki [145, Chap. IV℄.
64
Let us re all that the
ndimensional Lebesgue measure  is the restri tion of the ndimensional
outer Lebesgue measure  on
Rn
is dened as
n
A :# inf H I b ki " a ki : A k#1 i#1
algebra of Lebesgue measurable subsets of A # A S % A \ S for any S n .
on the
k k k k ℄ [ a 1 ; b 1 ℄,    ,[ a n ; b n ℄ ; k#1
R
Rn
. We all a set
A
a ki ¢b ki Ǳ
Rn
Lebesgue measurable if
26
Ë
1 Ba kground Generalities
ex eption of a set of Lebesguemeasure zero; referring to those holds, we will also say that it holds at almost all
x ò
x where this property x ò ).
(in abbreviation a.a.
:a.a. will mean that something holds for almost all elements.
The notation
1.3.a
Bo hner and Lebesgue spa es
L p ( ; S) we will denote the set of all Bo hner measurable65 fun tions66 u : Ù S su h that u L p ;S %, where
By
(
)
u L p
(
;S)
:#
p . 6 6 X u ( x ) S d x
> 6 6
1/
p
ess sup u(x) S xò
F
for
1 ¢ p % ;
for
p # % :
(1.36)
S is separable, Bo hner measurability is the same as strong meau : Ù S is alled " 1 strongly measurable if u ( A ) :# { x ò ; u ( x ) ò A } is Lebesgue measurable for any A ò S open with respe t to the strong topology. The set L p ( ; S), endowed with a pointLet us remark that, if
surability. Strong measurability has here the usual meaning:
wise addition and s alar multipli ation, is a linear spa e. Besides,  L p ( ;S) is a norm
L p ( ; S) whi h makes it a Bana h spa e, alled Bo hner spa e or, if S is innitep dimensional, a Lebesgue spa e. If S is separable, for 1 ¢ p %, L ( ; S) is separable too.67 Let us agree on the usual onvention that S will be omitted when equal to R. on
An important question is how to hara terize on retely the dual spa es. The natural duality pairing onsidered throughout this se tion will always ome from the
L2 spa es, whi h means :# P u1 (x)  u2 (x) dx, where u1  u2 will often abbreviate the duality pairing between S and S . If 1 p % and S is p a reexive Bana h spa e, then L ( ; S) is reexive. Using the Hölder inequality,68 it p
an be shown that the dual spa e is isometri ally isomorphi with L ( ; S ). This
hara terization of the dual spa e holds true also for p # 1.69 If S is not reexive, then p L p ( ; S) Ê Lw ( ; S ), whi h is the spa e of weakly measurable70 fun tions Ù S s alar produ t in
*
*
*
*
*
*
*
with the indi ated integrability;71 this is sometimes reers as a DunfordPettis theorem.
65
Bo hner's measurability means that
u is a.e. the limit of a sequen e of nitelyvalued measurable
fun tions.
66
As usual, we will not distinguished between fun tions that equal to ea h other a.e., so that, stri tly
speaking,
L p ( ; S) ontains lasses of equivalen e of su h fun tions.
67
See, e.g., Warga [791, Thm. I.5.18℄.
68
This is
P
u1 (x)

u2 (x)dx
¢
(P
p
p
u1 (x) S dx)1/p (P u2 (x) S dx)1"1/p ; *
f. e.g. Bourbaki [144,
Se t. IV.6.4℄ or Köthe [436, Se t. 14.10℄. Originally, Hölder [391℄ states it in a less symmetri al form for sums in pla e of integrals.
69
We refer, e.g., to Edwards [278, Thm. 8.20.5℄ or Gajewski et al. [342, Se t. IV.1.3℄.
70
A mapping
u* : Ù S* s ò S.
*
is alled weakly
measurable if
x
ÜÙ : Ù
measurable for any
71
We refer, e.g., to Edwards [278, Thm. 8.20.3℄ or [299, Thm. 12.2.4℄.
*
R
is Lebesgue
1.3 Fun tion and measure spa es
Moreover, for
s ; : : : sm ) #
( 1
1 p %
m 2 i #1 s i , the spa e
and
S #
Rm
Ë 27
equipped with the standard norm
L p ( ; S) is uniformly onvex.72
An important lass of nonlinear mappings from one Lebesgue (or Bo hner) spa e into another one onsists of the
xdependent superposition mappings:
N' (u) : L p ( ; S1 ) Ù L q ( ; S2 ) : u ÜÙ x ÜÙ '(x ; u(x))
;
(1.37)
S1 and S2 are separable Bana h spa es and ' : , S1 Ù S2 is a Carathéodory '(; s1 ) : Ù S2 is measurable for all s1 ò S1 and '(x ; ) : S1 Ù S2 is (strong,strong) ontinuous for a.a. x ò . The nonlinear mappings (1.37)
where
mapping, whi h means that
are alled Nemytski mappings.
Theorem 1.24 (Nemytski mappings).73 Let S ; S be separable Bana h spa es, ' : , S Ù S be a Carathéodory mapping, and 1 ¢ p %, 1 ¢ q ¢ %. Then the following 1
1
2
2
statements are equivalent to ea h other:
L p ( ; S1 ) into L q ( ; S2 ).
(i)
N'
maps
(ii)
N'
maps bounded subsets of
L p ( ; S1 ) onto bounded subsets of L q ( ; S2 ).
; a ò L q ( ) ;b ò R: '(x ; s ) S2 ¢ a(x) % b s Sp 1q . Moreover, if q #Ö %, then the above statements are also equivalent to p q (iv) N ' maps L ( ; S ) ontinuously into L ( ; S ). /
(iii)
1
1
1
2
In fa t, from the above general theorem, we will use only the impli ations74 (iii) (ii) and (iii)
á (iv).
á
S : Â± S is alled measurable if, for any open A S, S"1 (A) :# {x ò ; S(x) A #Ö } is measurable.75 An example of measurable
A multivalued mapping the set
multivalued mapping arises from level sets:
Theorem 1.25 (Measurable levelset mapping).76 Let ' : , S Ù R be a Carathéodory fun tion and let S : Â± S and : Ù R be measurable. Then the multivalued 72
This result is due to Clarkson [225℄, see also Adams [4, Corollary 2.29℄ or Kufner at al. [467, Re
mark 2.17.8℄.
73
For the full generality we refer to Lu
hetti and Patrone [499℄. If
S1
and
S2
are nitedimensional,
su h results an also be found, e.g., in Krasnoselski [440℄.
74
The former impli ation is just by Hölder's inequality and also the latter one has a relatively sim
Ù u in L q ( ; Rn ), then take subsequen es onverging a.e. on . Then, by ontinuity of ( x ; ) for a.a. x ò , N ( u k ) Ù N ( u ) a.e., and thus in measure, too. Due to the obviq ¢ 2q" (2 a q ( x ) % b q u ( x )q % b q u ( x )q ) for a.a. x ò , show that ous estimate ( x ; u k ) " ( x ; u ) k q { ( x ; u k ) " ( x ; u ) }kòN is equiabsolutely ontinuous sin e strongly onvergent sequen es are. Evenq tually ombine these two fa ts to get P ( x ; u k ) " ( x ; u ) Ù 0 and realize that, as the limit N ( u )
ple proof: Take
uk
1
is determined uniquely, eventually the whole sequen e onverges.
75
For this denition (possibly with
S
only omplete separable metri spa e) and further detailed
study of measurable multivalued mappings we refer to the monographs by Aubin and Frankowska [37, Chap. 8℄, Castaing and Valadier [188℄, or Deimling [256, Se t. 3℄.
76
Cf. Aubin and Frankowska [37, Theorems 8.2.9℄.
Ë
28
mapping
1 Ba kground Generalities
Â± S dened by x ÜÙ Lev S
(
x ); ( x ) ' ( x ; ) :# s ò S ( x );
'(x ; s) ¢ (x)
is measurable. Having a multivalued mapping
g(x) ò S(x) for any x ò .
S : Â± S, we say that g : Ù S is its sele tion if
Theorem 1.26 (Measurable sele tions).77
A multivalued mapping
S : Â± S
with
nonempty losed values is measurable if and only if there exists a sequen e { g k } k òN of its measurable sele tions
gk
su h that
S(x) # lS (U kòN g k (x)) for any x ò .
S : Â± S is measurable losedvalued, then also the multivalued mapping oS :
Â± S : x ÜÙ o(S(x)) is measurable.78 If S # Rn , one an onsider the following
If
modi ation of the Carathéodory theorem 1.2.4:
Theorem 1.27 (Carathéodory sele tions).79 Let S : Â± Rn be measurable nonempty n
losedvalued and g : Ù R be a measurable sele tion of o S . Then there are measurable sele tions g k ( x ) ò S ( x ) and measurable oe ients a k : Ù [0 ; 1℄ with k # 1; :::; n%1 su h that nk#% a k (x) # 1 and nk#% a k (x)g k (x) # g(x) for any x ò . 1
1
1
1
In the rest of this se tion, we will onne ourselves to the nitedimensional ase,
say
S :# Rm .
L p ( ; Rm ). Bounded sets in L p ( ; Rm ) are relatively weakly or weakly* ompa t provided 1 p % or p # %, respe tively. For p # 1 the situation is far more deli ate: Let us investigate the Lebesgue spa es
Theorem 1.28 (Weak L
1
 ompa tness). Let
M L1 ( ; Rm ) be bounded. Then the fol
lowing statements are equivalent to ea h other: (i)
M is relatively weakly ompa t in L1 ( ; Rm ),
(ii) the set
M is uniformly integrable, whi h means :
:" ¡ 0 ;K ò R% : (iii) the set
M
sup X uòM
{
x ò ; u ( x )£ K }
u(x)dx ¢ " ;
is equi ontinuous (or, more pre isely, equiabsolutely ontinuous) with
respe t to the Lebesgue measure, whi h means:
:" ¡ 0 ;Æ ¡ 0 :
77
sup sup X u(x) dx ¢ " ; u ò M A ¢ Æ A
This assertion is due to Castaing [186℄; see Aubin and Frankowska [37, Thm. 8.1.4℄ also for
other hara terization of measurability or also Castaing and Valadier [188, Se t. III.2℄, Deimling [256, Se t. 3.2℄ or Warga [791, Se t. 1.7℄.
78
Cf. Aubin and Frankowska [37, Thm. 8.2.2℄.
79
Cf. Aubin and Frankowska [37, Thm. 8.2.15℄.
1.3 Fun tion and measure spa es
(iv) there is a ontinuous fun tion
:
limaÙ% (a)/a # %) su h that:
R% Ù R%
Ë 29
with a superlinear growth (i.e.
sup X (u(x)) dx % : uòM
The points (ii) and (iii) are alled the DunfordPettis ompa tness riterion [274℄80 while the point (iv) is the de la ValléePoussin riterion [257℄.
L1 ( ) is losely related with the so alled 1 biting onvergen e by Cha on:81 A sequen e { u k } k òN L ( ) is said to onverge to B u ò L1 ( ) in the biting sense (then we will write u k Ù u), if there is a sequen e {A j }jòN su h that A j is measurable and A j A j %1 for any j ò N, limj Ù A j # , and u k Ù u for k Ù weakly in L1 (A j ) with j ò N arbitrary. The so alled Cha on biting 1 lemma [154℄ says that every bounded sequen e in L ( ) admits a subsequen e on1 verging in L ( ) in the biting sense. A bit more powerful version of the biting lemma The relatively weak ompa tness in
is the following:
Lemma 1.29 (Biting Lemma).82 Having a sequen e {u k }kòN bounded in L ( ), there are measurable A k su h that A k A k % for any k ò N, U k òN A k # , and su h that, 1
1
after taking possibly a subsequen e (denoted, for simpli ity, by the same indi es) the set
A k u k ; k ò N} is relatively weakly ompa t in L1 ( ), where A k : Ù {0; 1} denotes the hara teristi fun tion of the set A k .
{
Another important
L1 weak ompa tness
prin iple takes pla e for a.e. on
verging sequen es whi h have a ommon integrable majorant:
Theorem 1.30 (Lebesgue).83 Let {u k }kòN L ( ) be a sequen e su h that, for a.a. x ò , the sequen e { u k ( x )} k òN R onverges to some u ( x ) and u k ( x ) ¢ u ( x ) for some u ò L ( ). Then u lives in L ( ) and limkÙ PA u k (x) dx Ù PA u(x) dx for any A
measurable. In parti ular, the set { u k ; k ò N} is relatively weakly ompa t84 in L ( ). 1
0
1
0
1
1
It should be emphasized that the dual spa e to than
L
(
; Rm ) is substantially larger
L1 ( ; Rm ) and its elements an be identied with ertain measures. This leads us
to a few denitions from the measure theory. For simpli ity we will onne ourselves
80
See also, e.g., Della herie and Meyer [258, Chap.II, Theorems 19,22,25℄, Dunford and S hwartz [275,
Se t. IV.8℄, or Edwards [278, Se t. 4.21℄, where the relative sequential weak ompa tness in proved but, by the Eberlainmuljan theorem, it is equivalent to the relative weak ompa tness.
81
See Brooks and Cha on [154℄ or also Ball and Murat [67℄.
82
We refer to Valadier [774, Thm. 23℄.
83
See, e.g., Dunford and S hwartz [275, Corollary III.6.16℄ or Kolmogorov and Fomin [434, Se t. V.5.5℄.
Let us note that the linear hull of all hara teristi fun tions A with A measurable is dense in L ( ) Ê L1 ( )* , so that the sequen e {u k }, being bounded in L1 ( ), onverges weakly in L1 ( ) and,
84
as su h, it is relatively sequentially weakly ompa t, hen e by the Eberleinmuljan theorem relatively weakly ompa t, too.
30
Ë
1 Ba kground Generalities
to the s alar ase (i.e.
m # 1), the modi ation for the ve torial ase (i.e. m ¡ 1) being
obvious (ex ept the positive and the negative variations).
1.3.b
Spa es of measures
of subsets of an abstra t set M will be alled an algebra if ò , A ò á M \ A ò , and A1 ; A2 ò á A1 A2 ò . If also A i ò á UiòN A i ò , then will be alled a algebra. A fun tion Ù R is alled additive if ( A 1 A 2 ) # (A1 ) % (A2 ) provided A1 A2 # . If (U iòN A i ) # iòN (A i ) for any mutually disjoint A i ò , then is alled additive. For additive, we dene the variation I of by ( A ) # sup A A I ò M A i #1 ( A i ), where M ( A ) denotes the set of all nite 1
olle tions ( A 1 ; :::; A I ) of mutually disjoint A i ò for any i # 1 ; :::; I . Besides, the 1 1 % % positive variation is dened by ( A ) :# ( A )% ( A ), while the negative variation 2 2 " is dened by " (A) :# 12 (A) " 21 (A). The obvious identity # % " " is alled
A olle tion
(
;:::;
)
(
)
the Jordan de omposition.
Ù R with bounded variations will be deM; ), and its subset onsisting of additive set fun tions will be denoted
The set of all additive set fun tions noted by ba(
M; ). If M is additionally a topologi al spa e, then a set fun tion is alled regular if :A ò :" ¡ 0 ;A1 ; A2 ò : l(A1 ) A int(A2 ) and (A2 \ A1 ) ¢ ". If a set fun tion % " is additive, additive, or regular, then so are also all its variations , , and . In this ase, we an dene rba( M ; ) as the olle tion of all regular additive set fun tions Ù R with a bounded variation, and by r a(M; ) we denote its subset onsisting of additive set fun tions. The smallest algebra ontaining all open subsets of M
onsists just of all Borel subsets of M and, as su h, it will be alled the Borel algebra. n Often, M # will be a domain in R endowed not only by the Eu lidean topology, but also by the Lebesgue measure. Then another natural hoi e for the algebra is the set of all subsets of that are measurable with respe t to the Lebesgue measure.85 Then by vba( ; ) we denote the set of all additive set fun tions with bounded by a(
variations that vanish on sets having the Lebesgue measure zero. All the introdu ed spa es ba(
M; ), a(M; ), rba(M; ), r a(M; ), and vba(M; ) :#
are linear ve tor spa es whi h an be normed by means of the variation, i.e.
(M). This makes them Bana h spa es. Let us remark that additive set fun tions dened on a algebra are alled mea
sures, while the additive set fun tions are sometimes also alled nitely additive mea
M is a topologi al spa e and its Borel algebra, the measures from a(M; ) M; ) will be then addressed % % as Radon measures. If # with referring to the Jordan de omposition, is alled sures. If
will be alled Borel measures, while the elements of r a(
85
In fa t, this is the so alled Lebesgue extension of the Borel
sets of sets having the measure zero.
algebra, reated by adding all sub
1.3 Fun tion and measure spa es
Ë 31
algebra and a positive measure is the set of #(A) and dened as number of elements of A for nite subsets of M , otherwise as %. Moreover, a positive (nitely additive) measure will be alled a probability measure if ( M ) # 1. The onvex subsets of positive (resp. probability) measures will be % % % % % denoted by (resp. by 1 ), for example r a ( M ; ) or rba ( M ; ) (resp. r a1 ( M ; ) % or rba1 ( M ; )). An important example of a probability measure is the Dira measure Æ x supported at a point x ò M , whi h is dened for any subset A ò by a positive measure. An example of the all subsets of
M
and a so alled ounting measure, denoted by
Æ x (A) :#
1 0
if if
x ò A; x ò M \ A:
k
a i Æ x i of Dira 's measures with some x i ò M , a i £ 0, a i # 1, and k ò N is another example of a probability measure. We will all su h a measure k atomi . Borrowing a physi al terminology, su h measures are sometimes
A onvex ombination i #1
k i #1
alled mole ular, f. e.g. [501, Def. A.77℄.
Theorem 1.31 (Extreme probability measures).86 (i)
The Dira measures are extreme points in the set of all probability measures.
M is ompa t, then every extreme point in r a%1 (M; ) is of the form Æ x for some x ò M .
(ii) Conversely, if
B(M) we will denote the spa e of all bounded fun tions M Ù R endowed with u :# sup u(M), whi h makes it a Bana h spa e. If M bears 0 also a topology, say T , we denote by C ( M ) :# C ( M ) B ( M ) the linear subspa e of all By
the Chebyshev norm
ontinuous bounded fun tions endowed with the same norm, whi h makes it also a
M ; T ) is ompa t, then C0 (M) # C(M). If (M ; T ) is a lo ally ompa t spa e, then C 0 ( M ) will denote a losure of the subspa e of C ( M ) of fun tions with a ompa t support; the support of a fun tion g : M Ù S, denoted by supp(g), is dened by Bana h spa e. Let us note that, if (
supp(g) :# M \ UA ò T ; g(A) # 0: Likewise, having a measure
T ; (A)
86
# 0}. Let us note that the support is always a losed subset.
For the point (i) see, e.g., Köthe [436, Se t. 25.2℄. For
Lemma V.8.6℄ and realize that, if in
on M , we an dene its support by supp() :# M \ U{A ò
Æx
M ompa t see also Dunford and S hwartz [275, r a(M; ), it remains extreme
is an extreme point in unit ball of
r a%1 (M; ), too. The point (ii) follows from Dunford and S hwartz [275, Lemma V.8.5℄ if one realizes r a%1 (M; ) # o({Æ x ; x ò M}) and the set {Æ x ; x ò M},
that, by Proposition 1.16 and Remark 1.17, we have being a ontinuous image of the ompa t set
M , is weakly* ompa t.
Ë
32
1 Ba kground Generalities
Theorem 1.32 (Riesztype representations).87 Let M be a set with an algebra . Then: The dual spa e to B ( M ) is isometri ally isomorphi with ba( M ; ) provided is the
(i)
M. M is a normal topologi al spa e and the algebra generated by all losed subsets 0 of M , then the dual of C ( M ) is isometri ally isomorphi with rba( M ; ). If M is ompa t and the Borel algebra, C ( M ) is isometri ally isomorphi with r a(M; ). If M is lo ally ompa t and the Borel algebra, then C 0 ( M ) is isometri ally isomorphi with r a( M ; ). n If M is a measurable domain in R endowed with the Lebesgue measure and the algebra of all (Lebesgue) measurable subsets of M , then the dual spa e to L (M) is isometri ally isomorphi with vba( M ; ). algebra of all subsets of
(ii) If
(iii)
(iv)
(v)
*
*
In all ases, the isometri al isomorphism
ÜÙ g
*
, where
g
*
is the linear ontin
uous fun tional (as an element of the dual spa e in question) and
is the respe tive
measure, is given by the formula
g
*
; g :#
X g ( x ) (d x ) : M
The statements (iii) and (iv) are known as the Riesz representation theorems In parti ular, the Dira measure
Æx
[640℄.
an be understood as the linear ontinuous fun 
tional on ontinuous fun tions, whi h authorizes us to write Let us agree to use the shorthand notation omitting
Æ x (g) # g(x) for g ò C0 (M). (e.g. ba( ) in pla e of
ba( ; )) be ause the algebra will be always lear from a ontext. As vba( ) Ê L ( ) and L ( ) Ê L ( ) , we an see that vba( ) is (isomet
*
1
ri ally isomorphi with) the bidual of
*
L ( ). We saw in Se t. 1.2a that every Bana h 1
spa e an be anoni ally embedded into its bidual. A
epting this onvention, we will o
asionally not distinguish between integrable fun tions and the orresponding nitely additive measures (though sometimes the underlying integrable fun tions
L1 ( ) an be embedded also into measures understood as linear ontinuous fun tionals on C ( );
f. also Example 1.4.11 below. If a (nitely additive) measure possesses a density d ò L1 ( ), whi h means (A) # PA d (x) dx for any measurable A , then
will be addressed as densities of the measures in question). Alternatively,
has a ertain spe ial property, namely it is absolutely ontinuous with respe t to the Lebesgue measure, whi h means that
:" ¡ 0 ;Æ ¡ 0 :A measurable: A ¢ Æ âá
(A) ¢ ". Also the onverse assertion is true: every absolutely ontinuous measure L1 ( ). This is known as the RadonNikodým theo
possesses a density belonging to
87
For the parti ular points (i), (ii), (iii), and (v), we refer to Dunford and S hwartz [275, IV.5, Thm. 1
and Corollary 1℄, [275, IV.6, Thm. 2℄, [275, IV.6, Thm. 3℄, and [275, IV.8, Thm. 16℄, respe tively; for (v) f. also Yosida and Hewitt [803℄. For the point (iv), see, e.g., Edwards [278, Se ts. 4.3 and 4.10℄.
1.3 Fun tion and measure spa es
Ë 33
has a density d , we will use the notation (dx) # d (x) dx. Every òr a( ) admits89 a uniquely determined de omposition # 1 % 2 where 1 is absolutely ontinuous and 2 is singular (with respe t to the Lebesgue measure) in the sense that it is supported on some subset of having the Lebesgue measure zero; the splitting # 1 % 2 is alled the Lebesgue de omposition. n n Considering two Lebesgue measurable sets 1 R 1 and 2 R 2 , the identity
rem [578, 624℄.88 If measure
g(x1 ; x2 ) dx1 dx2 X
1 , 2 holds provided
#X
1
X g ( x 1 ; x 2 ) d x 2 d x 1
2
#X
2
X g ( x 1 ; x 2 ) d x 1 d x 2
1
g ò L1 ( 1 , 2 ) or provided one of the doubleintegral does exist and
is nite. This is known as the Fubini theorem90 [338℄.
1.3.
Spa es of smooth fun tions and Sobolev spa es
Ki,
Let us turn our attention to fun tions whi h enjoy some smoothness. Considering open and an in reasing sequen e of ompa t subsets
D( ) :#
k)
k
K i su h that #
U i òN
C K i ( ), where C K i ( ) denotes the spa e of all fun tions
Ù R whi h are ontinuous together with all their derivatives up to the order k and k whi h have the support ontained in K i . Ea h C K i :# V k òN C K i ( ) is endowed by the l
olle tion of seminorms { k K i } k òN with g k K i :# max1¢ l ¢ k x g C K i , whi h makes it a l Fré het spa e; here x is the ve tor of all partial derivatives of the order l . Then D( ) # U i òN C K is equipped with the nest topology that makes all the embeddings C K Ù i i D( ) ontinuous,91 whi h makes it a lo ally onvex spa e. More pre isely, D( ) is a Montel spa e.92 The elements of the dual spa e D( ) are alled distributions. we put
(
U i òN V k òN
( )
()
;
(
;
)
(
)
()
()
*
An important lass of fun tion spa es onsists of the Sobolev spa es [732℄, denoted by
Wk
;
p ( ;
Rm
) and dened, for
Wk
88
;
p
(
k ò N, by
; Rm ) :# u ò L p ( ; Rm );
k
x
u ò L p ( ; Rm,n
k
)
;
See also, e.g., Dunford and S hwartz [275, Se t. 3.10℄, Edwards [278, Se t. 4.15℄ or Halmos [374,
Se t. 31℄.
89
We refer, e.g., to Dunford and S hwartz [275, Thm. 3.4.14℄ or Edwards [278, Thm. 4.15.8℄.
90
See also, e.g., Halmos [374, Se t. 36℄, Kolmogorov and Fomin [434, Se t. 5.6.4℄, or Yosida [804,
Se t. 0.3℄. In fa t, Fubini's theorem holds in more general situations than Lebesgue measures on
91
This topology is alled the indu tive limit of the topologies on
C Ki
()
Rm
.
; see, e.g., Edwards [278, Se 
tions 5.1 and 6.3℄.
92
The Montel spa e is a barrelled spa e in whi h every bounded set is relatively ompa t; re all that
a lo ally onvex spa e is alled barrelled if every losed, balan ed, onvex, and absorbing subset is a neighbourhood of 0.
Ë
34
where D
1 Ba kground Generalities
k u denotes the set of all
kth
order partial derivatives of
Rm
u
understood in
m # ( ;R ) k u p 1/ p ) , whi h makes it a Bana h spa e. Likewise for m ( ;R ) L p ( ;Rm,n k ) 1; p m ) are separable and, Lebesgue spa es, for 1 ¢ p % the Sobolev spa es W ( ;
Wk
the distributional sense.93 The standard norm on
(
if
p
%
u L p
;
p ( ;
x
) is
u W k p ;
R
1 p %, they are uniformly onvex,94 hen e also reexive. Besides, for k £ 0 noninteger we dene Wk
;
p
(
; Rm ) :# u ò W
[
k ℄; p
(
; Rm );
k
X X
k. For k
where [ ℄ denotes the integer part of
u(x) " u(x )p dxdx %; n%p k" k x " x
(
[
(1.38)
℄)
noninteger,
Wk
;
p ( ;
Rm
) is alled the
ppower of)
SobolevSlobode ki spa e and the doubleintegral in (1.38) is alled (the Gagliardo's seminorm. They are Bana h spa es if normed by the norm
u W k p ;
We say that
(
;Rm )
the tra e operator
C( ; R
m)
:#
u
p W k p ( ;Rm ) [
℄;
%X
X
1 u(x) " u(x )p dxdx n % p k " k x " x
/
(
[
is the Lips hitz domain if its boundary
u ÜÙ u
Ù C( ; R
p
℄)
:
is Lips hitzian.95 Then
, onsidered lassi ally as a mapping
W 1 p ( ; Rm ) ;
m ), an be extended ontinuously to a linear, ontinuous, and
surje tive operator
u ÜÙ u : W 1 p ( ; Rm ) Ù W 1"1 ;
/
p; p
where the SobolevSlobode ki spa e on the boundary
ation of
so that
(
; Rm ) ;
(1.39)
is dened by the lo al re ti
is overed by Lips hitzian images of (
n " 1)dimensional domains
on whi h the former denition of SobolevSlobode ki spa es an be already used.96 The losed linear subspa e { Furthermore, we have
W 1"1
/
. 6 6
p:# >
u ò W 1 p ( ; Rm ); u # 0}
p; p (
;
; Rm ) L
p
(
; Rm ) with the notation
np " p n"p
for for
u ÜÙ u : W 1 p ( ; Rm ) Ù L p ;
(
W0
1;
p
(
; Rm ).
p n;
p # n; for p ¡ n :
an arbitrarily large real 6 6 F %
To summarize, we have
is denoted by
(1.40)
; Rm ).
k u/x1k1 ::: x knn with k1 % ::: % k n # k and k i £ 0 for any i # 1; :::; n is dened as an distribution su h that # ("1)k m ). for any g ò D( ;
93
For example, the distributional derivative
94
See Adams [4, Thm. 3.5℄.
95
It means that
R
an be divided into a nite number of overlapping parts, ea h of them being a
graph of a s alar Lips hitz fun tion on an open subset of
96
See, e.g., Adams [4℄ or Kufner, Fu£ík and John [467℄.
Rn"
1
.
1.3 Fun tion and measure spa es
Ë 35
Relations between various fun tion and measure spa es are often in the form of in lusions. Su h in lusions are always linear operators whi h an have some additional properties: the parti ular embedding is alled ontinuous, ompa t, dense, or
homeomorphi al if the orresponding linear operator is ontinuous, ompa t, have a dense range, or the inverse operator (restri ted on the range of the original operator) is ontinuous together with the original operator, respe tively. The following embedding theorems will be often used: The embedding
C( )
1 ¢ p % it is dense but not homeomorphi al p # % it is homeomorphi al but not dense. For 1 ¢ p ¢ q ¢ %, we have q p the ontinuous dense embedding L ( ) L ( ) (re all that we supposed bounded hen e %, otherwise this embedding would not hold). Neither of the mentioned L p ( ) is always ontinuous, and for
while for
embeddings is ompa t. On the other hand, it holds97
1 p
¡
1 q
"
k n
Wk
âá
;
q
(
) L p ( )
ompa tly
;
(1.41)
n is the dimension of Rn . If 1/p £ 1/q " k/n, then the embedding L p ( ) is generally only ontinuous provided kq n or kq # n # 1. Also, k q ( ) is ontinuously embedded into C ( ). Introdu ing for kq ¡ n £ 2 or for n # 1, W
re all that
W k ; q ( )
;
the notation (the so alled Sobolev exponent)
. 6 6
p :# > *
np n"p
for
an arbitrarily large real 6 6 F %
p n;
p # n; for p ¡ n ;
(1.42)
for
W 1 p ( ) L p ( ) or W 2 p ( ) L p ( ) with p :# (p ) . Also, e.g., we have u Ù Ü u : W 2 p ( ) Ù L p ( ) with p :# (p ), referring to the notation (1.40).
we an write
*
;
;
**
;
*
*
**
*
*
*
The embeddings an be transposed, resulting thus to relations between the
G1 G2 and denoting I : G1 Ù G2 , the adjoint operator I : G2 Ù G1 makes just the linear ontinuous fun tionals on G 2 . Let us distinguish
respe tive dual spa es. Having two fun tion (Bana h) spa es
*
the ontinuous embedding the restri tion on
G1
of
*
*
two typi al situations for su h
ontinuous embeddings:
where
97
Ti
! . onsistent, i.e. T2 !! !G 1 # T1 > dense, i.e. l I ( G 1 ) # G 2 F
denotes the norm topology of
type (C)
;
type (D)
:
(1.43)
G i , i # 1; 2. These two types more in details:
Re all that throughout the book we use the onvention
1/p :# 0 for p # %.
Ë
36
(
1 Ba kground Generalities
Su h embedding
)
I
*
: G Ù G *
*
2
1
I : G1 Ù G2
is homeomorphi al and then the adjoint operator
is surje tive be ause every linear ontinuous fun tional on
remains ontinuous also with respe t to the topology indu ed from
G1
and an
G2 by the HahnBana h theorem 1.11. I : G1 Ù G2 ontinuous and dense makes the adjoint
be then extended onto
d
(
G2
The embedding operator
)
I : G2 Ù G1 is inje tive (be ause two dierent linear ontinuous fun tionals on G 2 must have also dierent tra es on any dense subset, in parti ular on G 1 ). *
*
operator
*
Sometimes the above ases an appear simultaneously, whi h gives rise to the third situation when the ontinuous embedding is simultaneously onsistent and dense:
d
(
)
I : G Ù G is homeomorphi al and dense, then the adjoint : G Ù G is onetoone. Though I need not be a (weak*,weak*)
If the embedding operator
I
*
1
*
*
2
1
2
*
homeomorphism, it is the (weak*,weak*)homeomorphism if restri ted on a ball in
G2
*
duals
(whi h is weakly* ompa t). If
G1
*
and
G2
*
G1
G2
and
are normed spa es (so that the
are Bana h spa es), the inverse operator (
I )"1 *
is additionally
(strong,strong) ontinuous thanks to the openmapping theorem. In the situation (D) and thus also (CD), it is a ommon onvention to onsider bedded via images in
I
*
into
G1 .
G1
*
and then not to distinguish between elements of
G2
*
G2
*
em
and their
*
Example 1.33 (Intermediate subspa e G). Let :# ["1; 1℄ and let G be a linear spa e of fun tions g : Ù R whi h are ontinuous ex ept 0 where they posses unilateral limits; this means g " ò C(["1; 0℄) and g ò C([0; 1℄). We endow G with the (
1 ; 0)
(0 ; 1)
supremum norm. We have obviously
C(["1; 1℄) G L
"1; 1℄) ;
([
both embeddings98 being homeomorphi al but not dense, i.e. of the type (C) but not
G an be identied with the spa e of ertain measures, namely r a(["1 ; 0℄) , r a([0 ; 1℄). The relations between the dual spa e are obviously the surje tions: vba["1 ; 1℄ Ù G Ù r a["1 ; 1℄. Neither of these surje tions is invertible. For example, for any a ò R, the mapping aÆ 0" % (1" a ) Æ 0% : G Ù R dened by
(D). Again, the dual spa e
*
*
[
a Æ0" % (1" a) Æ0% ℄ (g) # a lim g(x) % (1" a) lim g(x) x ÷0
x ÿ0
G. Obviously, if g ò C(["1; 1℄), then aÆ0" % (1" a)Æ0% ℄ (g) # Æ0 (g) # g(0). In other words, the surje tion G Ù C(["1; 1℄) ,
forms a linear ontinuous fun tional on [
*
*
C(["1; 1℄) is a spa e of fun tions while L (["1; 1℄) onsists of equivalen e lasses of fun tions. Nevertheless, the embedding like C (["1 ; 1℄) L (["1 ; 1℄) has
98
To be pre ise, one should realize that
a good sense be ause two ontinuous fun tions, that are a.e. equal to ea h other, oin ide with ea h other; f. also Lang [475, Se t. VII.4℄.
1.3 Fun tion and measure spa es
Ë 37
C(["1; 1℄), sends the fun tional aÆ0" % (1"a)Æ0% to the Æ0 . Thus we saw the situation that the Dira measure is split onto a ontinuum of mutually dierent measures when the spa e of test fun tions C (["1 ; 1℄) is enlarged for G C (["1 ; 1℄). On the other hand, G is still a separable Bana h spa e, so that the weak* topology on bounded subsets of G is metrizable, ontrary to L (["1 ; 1℄). However, L (["1 ; 1℄) is no longer separable, whi h auses that the re
whi h is just the restri tion on Dira measure
*
sulting fun tionals likely annot be des ribed expli itly but merely their existen e an be laimed with help of the HahnBana h theorem (and thus of the axiom of hoi e whi h is involved nontrivially in the HahnBana h theorem).99
Example 1.34.
D( ), L p ( ) an be transposed for the respe tive dual spa es. We suppose bounded and 1 ¢ p %. The relations are summarized
C( ), L
(
For an illustration, let us realize how the interrelations between
), G
from Example 1.33, and
by the following diagram. The arrows are marked either by (C) or by (D), referring thus to the above lassi ation.
(D) ✲ (C) ✲ G (C) ✲ L ( ) D( ) C( ) ❳❳❳ ✘✘ (D) (D) ✘✘✘✘ (D) ❳❳❳❳❅❅ (D) ✠✘ ✾✘ ③❘ ❳ L p ( )
The transposed diagram is the following (the des ription
sur and inj indi ates re
spe tively the surje tivity or inje tivity of the mapping orresponding to the parti ular arrow):
INJ. r a( ) ✛SUR. SUR. vba( ) D( )* ✛ G* ✛ ②❳❳ ❳ ✿ ✘✘ ■ INJ. ✒ INJ.✘✘✘✘ INJ. ❳❳❳❳❅ INJ. ✘ ❳❅ ✘ L p ( )
The relations between the involved spa es are a
omplished by the observation that the rst diagram is onne ted with the se ond one be ause we have always the em
L p ( ) Ù vba( ). For p £ 2, we have even stronger onne tion be ause there p p is the embedding L ( ) Ù L ( ).
bedding
Remark 1.35 (Insu ien y of the on ept of sequen es). The weak* topology on vba( ) Ê L ( ) is not metrizable even if restri ted on bounded subsets, whi h is 1
**
related with the fa t that
L
(
)
is not separable. Besides, this is an example of a
situation where sequen es are not a satisfa tory tool. Namely, no element from the
vba( ) \ L ( ) an be attained (with respe t to the weak* topology) by a sequen e from L ( ), though L ( ) is dense in vba( ). Indeed, if it were possible, 1
remainder
1
1
su h a net would be weakly Cau hy in
99
L1 ( ) be ause the tra e of the weak* topology
Cf. also the example by Lang [475, Se t. VII.4℄.
Ë
38 in
1 Ba kground Generalities
vba( ) oin ides with the weak topology in L ( ). However, the limit of su h a se1
quen e must live in
L ( ) be ause L ( ) is sequentially weakly omplete.100 Anyhow, 1
1
insu ien y of sequen es (and ne essity of the on ept MooreSmith onvergen e) is onsidered as too fargoing mathemati al abstra tion whi h dramati ally looses
onstru tivity and is attempted to be avoided in appli ations.
Some dierential and integral equations
1.4
In this se tion we will briey summarize some basi fa ts about sele ted lasses of dierential and integral equations whi h we will need in the examples of Chapter 4.
Ordinary dierential and dierentialalgebrai equations
1.4.a
We will start with the initialvalue problem for a (system of) ordinary dierential equa
tions (ODE) we will also say for a nitedimensional dynami al system:
dy # f(t ; y) with t ò I and y(0) # y dt n n n with a Carathéodory mapping f : I , R Ù R with I :# [0 ; T ℄ and y ò R .
(1.44)
0
0
The main ingredient for estimation of evolution systems in general is the so alled
Gronwall inequality,101 whi h we will also often use. In the general form, this inequality says that, for all
t £ 0, it holds y(t) ¢
whenever we know that
C
t
% X b()e" P0 a # (
)
d#
0
t
d eP0 a (
)
d
(1.45)
t
y(t) ¢ C % P0 a()y() % b() d for some a ; b £ 0 integrable.
Classi al existen e and uniqueness results are the following:
Proposition 1.36 (Ordinary dierential equation).102 Let 1 ¢ p ¢ % and the Caran n p théodory mapping f : I ,R Ù R satisfy f ( t ; r ) ¢ a p ( t )(1 % r ) with some a p ò L ( I ). p n Then (1.44) possesses a solution y ò W ( I ; R ). This solution is unique provided f ( t ; r ) " f ( t ; r ) ¢ a ( t ) r " r for some a ò L ( I ). 1;
1
2
1
1
2
1
1
¡ 0 with T/ ò N, let us dene the approximate solution y ò W 1 (I; Rn ) su h that, for any k # 1 ; :::; T / , the restri tion y k "1 k is ane and, denoting ;
Proof. For
[(
100
)
;
℄
For this nontrivial fa t the reader is referred, e.g., to Dunford and S hwartz [275, Se t. IV.8℄ or
Edwards [278, Thm. 4.21.4℄.
101
In the general form presented here, whi h an be found, e.g., in Mordukhovi h [550, Se t. B1℄, it
is also alled the BellmanGronwall inequality. 1 In fa t, for p # 1, it su es to assume f ( ; 0) ò L ( I ; f(t ; ), we an see f(t ; r) ¢ a1 (t)(1 % r), too.
102 of
Rn
) be ause then, by the Lips hitz ontinuity
1.4 Some dierential and integral equations
Ë 39
y k # y (k), given by the re ursive formula holds103 k
y k # y k"1 % X
(
k "1)
f(t ; y k"1 ) dt ;
k # 1; :::; T/, starting for k # 1 with y0 # y0 . Due to the estimate y k " y k"1 / ¢ k "1 ) P k (1% y a (t) dt with a0 ò L1 (I), we an see that y is bounded in L (I; Rn ) k "1 0
for
(
)
. Sin e even a0 ò L p (I), we an also see that ddt y is bounded in L p (I; Rn ) uniformly with respe t to . Sin e p ¡ 1, we an take a subsequen e and 1 p n some y ò W ( I ; R ) su h that, for Ù 0, it holds
independently of
;
y Ù y
weakly* in
W 1 p (I; Rn ) ; ;
1 p % it is the weak onvergen e while for p # 1 this onvergen e BV(I; Rn ) but, as we later show that ddt y # Nf (y) ò L (I; Rn ), the limit y n k " for t ò [( k "1) ; k ) and n ( I ; R ) by y ( t ) # y belongs to W ( I ; R ). Dening y ò L k # 1; :::; T/, we an see that
in fa t, for
1
is in rather in
1;1
y Ù y be ause
strongly in
L q (I; Rn )
for any
q %
y Ù y strongly in L p (I; Rn ) and be ause of the al ulus
p
y "y L p I;Rn # (
)
#
T/
k !! k t !!( y " y k "1 ) HX !! ( k "1) k #1 T/ !!! k k"1 !!!p H !y "y ! # p%1 k#1 !! !!
" (k"1) !!!p
!! !
dt
p """ dy """ p # O ( p ) " " p%1 "" dt "" L p I;Rn (
)
I; Rn ). Passing to the limit in the obvious identity # Nf (y ) and using the ontinuity of the Nemytski mapping Nf : L p (I; Rn ) Ù
and be ause of the bound of
d dt y
1
y in L
(
L p (I; Rn ), we get ddt y # Nf (y), whi h just means that y solves (1.44); note that y(0) # y0 be ause y (0) # y 0 and be ause of the weak ontinuity of the tra e operator y ÜÙ y (0) : W 1 p (I; Rn ) Ù Rn . Supposing now that (1.44) admits two solutions y 1 and y 2 , we get by subtra tion and multipli ation by y 1 " y 2 the estimate ;
1d y " y 2 dt 1
2
2
¢ Nf (y ) " Nf (y
y1 # y2 a # a1 , and y # y1 "y2 2 .
from whi h we get
103
1
2 )
y " y2 ¢ a1 y1 " y2 2 ;
 1
by the Gronwall inequality (1.45) used for
C # 0, b # 0,
In other words, we use the so alled (expli it) Euler formula with an equidistant partition of the
time interval
I.
40
Ë
1 Ba kground Generalities
A useful generalization of the initialvalue problem for ordinary dierential equations (1.44) is towards dierentialalgebrai 104 equations (DAE) in the so alled semi
expli it form.105 Conning ourselves again to nitedimensional ases, it reads as
dy # f(t ; y; w) ; y(0) # y dt 0 # g(t ; y; w) with Carathéodory mappings
y0 ò R
and with
n . Now
f : I,
y(t) ò R
m and
r
(1.46a)
0
(1.46b)
Rn , Rm Ù Rn and g : I , Rn , Rm Ù Rm , w t ò Rm are unknown ve tors of slow and ( )
y, we will use v a pla eholder for values of w. Saying that (1.46) has a (dierential) index k means that we need to dierentiate the algebrai part (1.46b) in time ( k "1)times to obtain
fast variables, respe tively. Like
being a pla eholder for values of
the underlying system of ordinary dierential equations (ODE) like (1.44). The simplest DAEs with index 1 arises when the algebrai part (1.46b) admits an impli it fun tion
w in the sense:
;w : I , Rn Ù Rm :
g(t ; r; v) # 0 ã v # w(t ; r) :
(1.47)
This assumption is to be veried in ea h parti ular ase in on rete appli ations. Then the so alled underlying ODE (1.44) takes solution
y
f
as
of this underlying ODE, the pair (
f w : (t ; r) ÜÙ f(t ; r; w (t ; r)). Having a y; w) with w # w(t ; y) solves the DAE
(1.46). The ondition (1.47) often annot be fullled be ause the DAEs in question have an index higher than 1. For index2 DAEs, we will assume satisfying
g smooth (of the C1  lass),
gv # 0 ;
this means that
(1.48)
g depends only on t and y. By dierentiation of the algebrai equation
(1.46b) on e in time and using also the dierential equation (1.46a), one gets
0#
d dy g(t ; y) # g t (t ; y) % g r (t ; y) # g t (t ; y) % g r (t ; y)f(t ; y; w) : dt dt
(1.49)
The analog of the assumption (1.47) now reads as
;w : I , Rn Ù Rm : Then, using this repla ing
104
f
g t % g r f (t ; r; v) # 0
ã
v # w(t ; r) :
w, the underlying ODE is to be onstru ted and used as before when f w. Now, the initial ondition y is to be ompatible with the
in (1.44) by
0
The adje tive algebrai in the ontext of DAEs does not refer to any algebra, just wants to high
light that
wvariable is derivative free. Sometimes they are also alled singular systems, referring that E ddt y # f(t ; y) with a matrix
(1.46) an alternatively be understood as a generalization of (1.44) towards
E whi h an be singular. 105
(1.50)
The adje tive semiexpli it refers to that
d dt y o
urs expli itly in the dierential part.
1.4 Some dierential and integral equations
Ë 41
algebrai onstraint, namely
g(0; y0 ) # 0 :
(1.51)
Higherindex DAEs be ome more umbersome. Let us still present index3 DAEs, whi h has importan e in some appli ations, f. Remark 4.73. Then of
g is to be assumed
C2  lass satisfying, in addition to (1.48), also g r (t ; r)f v (t ; r; v) # 0 :
(1.52)
By dierentiating the algebrai equation (1.46b) twi e in time and using also the differential equation (1.46a) and the stru tural restri tions (1.48) and (1.52) in order to eliminate expli it dependen e of
0#
w on ddwt , one obtains
d d g(t ; y) # g t (t ; y) % g r (t ; y)f(t ; y; w) dt dt dy dy # g tt (t ; y) % g tr (t ; y) % g tr (t ; y) % g rr (t ; y) f(t ; y; w) dt dt dy dw % g r (t ; y) f t (t ; y; w) % f r (t ; y; w) % f v (t ; y; w) dt dt # g tt (t ; y) % g rr (t ; r)f (t ; r; v) % g r (t ; y)f r (t ; y; w)f(t ; y; w) % 2g tr (t ; y)f(t ; y; w) % g r (t ; y)f t (t ; y; w) : (1.53) 2
2
2
Let us note that (1.48) now implies also
g yv # 0. Instead of (1.50), we now assume
;w : I , Rn Ù Rm : g tt % g rr f 2 % g r f r f % 2g tr f % g r f t (t ; r; v) # 0
Again, using this
ã
v # w(t ; r) :
(1.54)
w, the underlying ODE is to be onstru ted and used as before. More
over, it is also natural (and to some extent ne essary) to assume the initial velo ity
d dt y(0) ompatible with the algebrai onstraint (1.46b), i.e. .
g t (0; y0 ) # "g r (0; y0 )y0 ;
where
y0 :# f(0; y0 ; v) ; v ò Rm : .
independent of v . The imporg r (0; y0 )y 0 in (1.55) does not depend on r and v . be ause of the orthogonality (1.52), though y 0 itself may depend on v as indi ated in
Here we used the assumption (1.48) implying tant fa t is that the righthand side
.
gt
(1.55)
and
gr
(1.55). Let us summarize the above manipulations towards using Proposition 1.36:
Proposition 1.37 (Dierentialalgebrai systems).106 Let (1.47), or (1.48) with (1.50) or
with (1.52) and (1.54) hold. Moreover, in the latter two ases, let y 0 be ompatible with the
f w ts with the assumptions y ò W 1;1 (I; m ) to ddt y # m ) sin e w ( t ) # w( t ; y ( t )). f(t ; y; w(t ; y)) with y(0) # y0 , from whi h we then get w ò L (I;
106
Note that the assumptions on
of Proposition 1.36 with
p
# 1,
f
and
w
are just devised so that
whi h then yields a unique solution
R
R
42
Ë
1 Ba kground Generalities
algebrai part in the sense (1.51) or (1.55), respe tively. Let also
w : I ,Rn Ù Rm from
(1.47), or (1.50), or (1.54) be a Carathéodory mapping uniformly Lips hitz ontinuous in the sense
w(t ; r)"w(t ; r ) ¢ C(1 % r" r ) for some C ò R with w(; 0) ò L (I; Rn )
f : I ,Rn ,Rm Ù Rn satisfy f(; 0; 0) ò L1 (I; Rn ) and f(t ; r; v)" f(t ; r ; v )) ¢ a1 (t)(1 % r" r % w" v ) with some a1 ò L1 (I). Then the initialvalue problem (1.46) has a 1 1 m m unique solution ( y; w ) ò W (I; R ) , L ( I ; R ).
and and let
;
Of ourse, if (1.46b) ontains
m ¡ 1 equations, the index may be dierent in dier
ent equations and the above al ulations should then be ombined. The presen e of the algebrai onstraint (1.46b) may bring di ulties in numeri al solutions of DAEs in omparison with ODEs, and may exhibit some hidden onstraints in parti ular in the ontext of optimal ontrol of systems governed by DAEs, f. Se tion 4.3.g. Another useful generalization of the initialvalue problem for ordinary dierential equations (1.44) is towards innitedimensional dynami al systems, i.e. (1.44) with a
f : I , V Ù V and y0 ò H with V being a separable reexive Bana h spa e and H V a separable Hilbert spa e; i.e. here f ( t ; ) : V Ù V strongly
ontinuous for a.a. t ò I and f ( ; v ) : I Ù V Bo hner measurable for all v ò V . To be a bit more spe i , instead of f ( t ; v ) we will onsider f ( t ; v ) " A ( v ) with A : V Ù V , so *
Carathéodory mapping
*
*
*
that (1.44) will take the form
dy % A(y) # f(t ; y) dt Moreover, we assume that
with
tòI
and
y(0) # y0 :
(1.56)
H is identied with its own dual and the embedding V H
is ontinuous and dense; i.e. the embedding of type (D) so that the adjoint mapping
V H V . Importantly, the restri tion of the V on H is the s alar produ t (; ) on H and we have *
is inje tive. Therefore between
V
*
and
;
duality the abstra t
bypart integration formula
t dy ; y½ X ¼ 0
dt
so that in parti ular for
%¼
dy ; y½ dt # (y(t); y (t)) " (y(0); y (0)) ; dt
t
y # y we have P0 < ddyt ; y > dt #
yt
1
2
( )
2
H
"
1 2
y 0) 2H . To devise
(
the abstra t s heme optimally, we expe t to have some Bana h spa e
L V;H ; p ontain1" 
I; H) for whi h the interpolation  LV H p ¢ C  L p holds for some C ò R and 0 1.
ing
L p (I; V) L
(1.57)
(
;
;
I V)
( ;
L
I H)
( ;
There are several te hniques to handle this evolution problem in its various generality. For simpli ity, having in mind appli ation to paraboli partial dierential equations, we onne ourselves to the monotoni ity te hnique.
Proposition 1.38 (Solutions to abstra t dynami al system). Let the embedding V H V V for some Bana h spa e V , A : V Ù V be ontinuous and f : I ,V Ù V be a Carathéodory mapping bounded in the
be dense and ompa t as well as the embedding *
*
1.4 Some dierential and integral equations
Ë 43
sense107
p "1
A(v)V ¢ C( v H )1% v V *
for some
and
p "1
f t ; v) V ¢ p (t) % C( v H ) v V
(
*
R Ù R ontinuous, and let further A " f t ;
1 p %, C :
(
)
(1.58a)
:VÙV
*
be
semi oer ive in the sense
A ( v )
" f(t ; v); v £ vpV " p (t)vV " (t) v H
2
1
(1.58b)
q ò L q (I), and A(u) # A(u ; u) with A(; v) : V Ù V ontinuous with some Bana h spa e V into whi h V is embedded ompa tly and A( u ; ) : V Ù V ontinuous *
with some
*
and uniformly semimonotone in the sense
A( u ; v )
" A(u ; v ); v" v £ v" v pV " v" v H / 2
(1.58 )
¡ 0 for some seminorm  V on V satisfying v V ¢ C(vV % v H ) for some C %, N f maps bounded sets in L p (I; V) L (I; H) into bounded sets in Lp V H p, p 1 p and y 0 ò H. Then (1.56) has a solution y ò L ( I ; V ) W ( I ; V ) in the sense that d y % A(y) # f(y) holds a.e. on I in V and y(0) # y holds108 in H . Moreover, if also 0 dt with some
;
;
;
*
*
; ¡ 0 ò L (I) :v; v ò V : 1
1
A ( v )
" f(t ; v) " A( v ) % f(t ; v ); v" v % (t)% v pV % v pV v" v H / £ 0 ; 2
1
(1.58d)
then this solution is unique and depends ontinuously on the data in the sense that, for
f(t ; y) # f0 (t ; y) % f1 (t), the mapping f1 ÜÙ y is ontinuous from L p (I; V*) Ù L p (I; V) L (I; H).
Sket h of the proof. We use the approximation by FaedoGalerkin's method exploiting theory of ordinary dierential equations as in Proposition 1.36. Let us take a sequen e
V1 V2 V3 ::: V whose union is dense in V y0 k ò V k su h that y0 k Ù y0 in H . For k ò N, let us dene109 the approximate 1 p solution y k ò W ( I ; V k ) su h that y k (0 ; ) # y 0 k . 1 d 2 The test of the approximate solutions by y k is legitimate and gives y k H % 2 dt < A ( y k ) ; y k > # < f ( t ; y k ) ; y k >. Then, using (1.58b) and the growth (1.58a) of f and the of nitedimensional subspa es
and
;
;
;
107
;
In fa t, (1.58a) an be generalized by allowing
108
f
to have also a omponent admitting the bound
¢ (t)(1 % v H ) with some ò L (I). p p ( I ; V ) is embedded into C ( I ; H) so that the initial ondition A tually, the spa e L ( I ; V ) W
f t ; v) H
(
1
1;
*
has indeed a good sense.
109
Considering a base {
tion
yk
v i }i#1;:::;k
of
Vk
and the ansatz
y k (t)
#
k i#1
is determined by a system of ordinary dierential equations
f t ; kj#1 j (t)v j ); v i > with i
< (
i (t)v i , the approximate solu% #
d dt i
# 1; :::; k for the oe ients i , so that the existen e of our approximate
solutions an be laimed by Proposition 1.36 rst lo ally in time and then by ontinuation using the
L (I)apriori estimates.
44
Ë
1 Ba kground Generalities
Young inequality, the estimate
1d y % y k pV ¢ y k H % p y k V : 2 dt k H 2
2
1
y 0 # y0 H %, I; H) L p (I; V). By omparison and using (1.58a), for d any k £ l we obtain a uniform bound of dt y k in seminorms
Using the Young and the Gronwall inequalities together with k ( ) H
y k in L
we obtain the bound of

(
l :#
sup
:a.a. tòI: v(t)òV l v L I;H L p I;V ¢1
(
)
(
T X  ; v ( t ) d t :
(1.59)
0
)
Then, by Bana h's sele tion prin iple (Theorem 1.9), we take a subsequen e
y
onverging weakly* in
L
(
I; H) L p (I; V).
Using ompa tness of
V V
y ki Ù
and the
AubinLions theorem110 about the ompa t embedding
L p (I; V) W 1 1 (I; Vl s ) L p (I; V ) ;
ompa tly
Vl s ,111 we still have y k Ù y strongly in L p (I; V ). By the interpolation with the boundedness in L ( I ; H), we have y k Ù y strongly also for any Hausdor lo ally onvex spa e
in
Lp V ;H ; p . Furthermore, we prove strong onvergen e
strategy (1.34). More spe i ally, taking
yl
y ki Ù y
by modifying the abstra t
y
an approximation of
valued in
Vl
and
using (1.58 ), we estimate
1d 1 p y l "y k i H % y l "y k i V ¢ A( y k i ; y l )" A ( y k i ) ; y l " y k i % y l " y k i H 2 dt 1 # A(y k i ; y l ); y l "y k i " f(y k i ); y l "y k i % y l "y k i H ; 2
2
2
(1.60)
tdependen e of f , y k , et . for notational simpli ity. We an onsider y l Ù y strongly in L p (I; V) L (I; H). We then obtain the strong onvergen e y k i Ù y in L p (I; V) L (I; H) by the Gronwall inequality and by the onvergen e p # 0
i Ù
fy
sin e { ( k i )} i òN is bounded in
110 111
Lp*
V ;H ; p
and
y l "y k i Ù 0 strongly Lp V H
;
;
p.
See J.P. Aubin [34℄ and J.L. Lions [495, Chap. 1, Thm. 5.1℄. This is, in fa t, a bit te hni al generalization of the usual AubinLions theorem [659℄ tted for the
d dt y k holds only in the seminorms (1.59), whi h yields the Vl s . Alternatively, one an use the Bana h spa e V * in the position of Vl s and a Hahnd Bana h extension of dt y k , f. [685, Se t.8.4℄.
Galerkin approximation, as the estimate of topology of
1.4 Some dierential and integral equations
Having now the strong onvergen e
y ki Ù y
Ë 45
proved, the limit passage in the
Galerkin approximation towards (1.56) is then easy.
y1 and y2 and using (1.58d) and the Gronwall d inequality, we an see uniqueness. More spe i ally, testing the dieren e of dt y i % A(y i ) # f(y1 ), i # 1; 2, by y1 " y2 #: y12 and using (1.58d), we obtain the estimate Eventually, omparing two solutions
1d y 2 dt from whi h we obtain
2
12
H
¢ ( t ) % y 1
1
p V
% y
2
p 2 V y 12 H /
y12 # 0 by the Gronwall inequality, using also y12 t#0 # 0. Moref1 ÜÙ y is by the uniform monotoni ity
over, the laimed ontinuity of the mapping
(1.58 ), just repli ating the arguments for the strong onvergen e of the Galerkin approximation above.
Remark 1.39 (Weak solutions). I
Instead of the two equations in (1.56), the former one
holding a.e. on , one an require
T X A ( y )" f ( y ) ; v 0
for any
y ò L p (I; V) Cw (I; H) to satisfy the integral identity
" ¼y;
v ò L p (I; V) W 1 p (I; V ;
*
dv ½ d t % y ( T ) ; v ( T ) # y ; v (0) dt
). If also
0
d dt y
ò L p (I; V
*
), it is equivalent to the
lassi al solution to the initialvalue problem (1.56) laimed in Proposition 1.38.
Remark 1.40 (Abstra t dierentialalgebrai systems).
These two generalizations of
(1.44) an be ombined and thus one gets innitedimensional dierentialalgebrai systems, f. Remark 4.114.
1.4.b
Partial dierential equations of ellipti type
Another type of (systems of rst ase
m) dierential equations ontains partial derivatives. The
is of the ellipti type.112 Su h type of equations des ribes stationary (or
steadystate) spatiallydistributedparameter systems on a spatial domain and need also an appropriate boundary onditions. We will onne ourselves to the ase of Robintype (sometimes also alled Newton or Fouriertype) boundaryvalue problem for a 2ndorder systems of ellipti dierential equations in the divergen e form. So we will onsider the problem
"div a(x ; y; x y) % (x ; y ; x y) # 0 n (x)  a(x ; y; x y) % b(x ; y) # 0
112
; on ;
on
§
(1.61)
Here, readers are re ommended to the monographs e.g. [495, 568, 685℄ for more details and results
in more general situations.
Ë
46
where
1 Ba kground Generalities
is a bounded domain in
Rn with a Lips hitz boundary
notes the unit outward normal113 to the boundary
tion is pres ribed. The notation x denotes the spatial gradient ( x 1 ;
n
: ,R
R
m,n
Ù
R
m and
b :
,R Ù m
R
de
x n ) while m,n Ù m,n ,
:::;
a : ,Rm , R
div is the divergen e of a ve tor, i.e. i #1 x i () i . Here,
m,
n
and where
where the Robin boundary ondi
R
m are Carathéodory mappings repre
senting a ondu tivity or elasti ity oe ients, distributed sour es or for es, and a boundary ux or a for e tra tion, respe tively, depending on parti ular appli ations. As the lassi al (= pointwise) understanding of the problem (1.61) is not natural from both mathemati al and physi al reasons, the standard understanding of (1.61) is in the sense of distributions, whi h leads to the notion of a so alled weak solution. The weak formulation arises by multiplying the equation in (1.61) by some test fun tion by integration over
y,
, by the applying Green's formula, i.e. X y
whi h holds for any

div A % A : x y dx #
y : Ù
Rm and A
X
A : (y n ) dS
Rm,n smooth enough, and eventu
: Ù
ally by substitution of the onormal derivative from the boundary ondition in (1.61), whi h eventually yields the identity
X a ( y; x y ) : x y
where
:
% (y ; x y )  y dx % X b(y)  y dS # 0 ;
(1.62)
and  denotes the summation over two or one indi es, respe tively. For
notational simpli ity, we omit the expli it
R
xdependen e in a, , and b. We say that
y: Ù m is a weak solution to (1.61) if the integral identity (1.62) is fullled for any y ò W 1; p ( ; m ). Considering a suitable polynomiallikegrowth exponent 1 p
R
% of the data, we will seek a solution in the Sobolev spa e W y ò W p ( ; Rm ) in (1.62).
1;
p ( ;
Rm
) and use
1;
y; y ò b. First,
To guarantee the integral identity (1.62) to have a good sense for any
W 1 p ( ; Rm ), ;
one must impose a ertain onditions on the data
a, ,
and
all of them will be assumed Carathéodory mappings so that all terms under the integrals in (1.62) will be measurable. Moreover, the integrability of these terms will be respe tively guaranteed by the following growth onditions:
; ò L p ( ) :(x ; r ; ) ò , Rm , Rm,n :
; ò L p " (
; òL p (
*
)
(
)
a(x ; r ; ) ¢
:(x ; r) ò , R :
b(x ; r) ¢
R R
") ( ) :(x ; r; ) ò , m , m,n
:
x ; r; ) ¢
(
113
x % Crp
Let us note that this normal does exist a.e. on
*
( )
/
p
"
% Cp" ; 1
x % Crp ""1 ;
( )
(1.63b)
x % Crp ""1 %Cp *
( )
be ause
(1.63a)
/(
p
*
") ;
(1.63 )
is Lips hitzian, whi h means that
be overed by a nite number of graphs of Lips hitz fun tions on domains in
Rn"
1
.
an
1.4 Some dierential and integral equations
Ë 47
p , p, and p are from (1.35), (1.40), and (1.42), respe tively, C ò R, and 0 ¢ min(p ; p) " 1 is arbitrarily small. Let us a
ept the notational short ut that, for p ¡ n, the terms r% o
urring in (1.63b, ) are to be understood su h that b(x ; ), and ( x ;  ; ) may have an arbitrary fast growth if r Ù .
where
*
*
In view of Theorem 1.24, the growth onditions (1.63) are designed so that respe tively
Na : W 1; p ( ; Rm ),L p ( ; Rn ) Ù L p ( ; Rn )
,
is (weak strong,strong) ontinuous
y ÜÙ Nb (y ) : W 1 p ( ; Rm ) Ù L p " ;
(
)
(
; Rm )
is ontinuous
;
(1.64a)
;
(1.64b)
N : W 1; p ( ; Rm ),L p ( ; Rm,n ) Ù L(p ") ( ; Rm ) is ontinuous : *
(1.64 )
y; v ò W 1 p ( ; Rm ), the integrands a(y; x y) : x y and (y; x y)  y 1 1 o
urring in (1.62) belong to L ( ) while b ( y )  y belongs to L ( ). ;
In parti ular, for
Proposition 1.41 (Ellipti equations: existen e and uniqueness). Let 1 p % and the following uniformmonotoni ity114
oer ivity onditions are valid for some ¡ 0:
for some
:
x ò : r ò Rm ; ; ò Rm,n :
a.a.
X a ( y; x y ) : x y
"a(x ; r; ) : ( " ) £ " p ; 1 b(y)  y dS £ y W 1 p Rm " :
a ( x ; r;
% (y; x y)  y dx % X
(1.63) be valid and the overall
)
1/
;
(
;
)
(1.65a) (1.65b)
Then the boundary value problem (1.61) possesses at least one weak solution
W 1 p ( ; Rm ). Moreover, if the overall stri t monotoni ity hold, i.e.
y ò
;
:y; y ò W
1;
p
(
; R m ) ; y #Ö y :
X a ( y; x y )
" a( y ; x y ) : x (y" y )
% (y; x y) " ( y ; x y )  (y" y ) dx % X b(y) " b( y )  (y" y ) dS ¡ 0;
(1.66)
then the problem (1.61) possesses just one weak solution.
Sket h of the proof. We will use Proposition 1.23. The monotone mapping
W 1 p ( ; Rm ) Ù W 1 p ( ; Rm ) ;
A( y; z ) ;
y # X a(y; x z) : x y dx
(1.67a)
with
while the lowerorder ompa t part
A 2 ( y ) ;
114
:
is now determined by the inte
*
gral identity
A1
A1 (y) # A(y; y)
;
A2 is given by
y # X (y; x y)  y dx % X b(y)  y dS :
(1.67b)
When weakening (1.65a) to stri t monotoni ity, the proof is more involved, f. e.g. [495℄ or [685,
Lemma 2.32℄.
Ë
48
1 Ba kground Generalities
y; ) learly follows from (1.65a). The growth onditions A and A2 . Moreover, due to the ompa t embed1 p p " ( ) and hen e also of L p " ( ) W 1 p ( ) , we obtain the ding W ( ) L
ompa tness of A 2 . Thus, using still the oer ivity (1.65b) of A 1 % A 2 , the existen e of
The uniform monotoni ity of A(
(1.63) imply the required ontinuity of *
;
(
*
)
;
*
solution is due to Proposition 1.23(i). When (1.66) holds, we onsider two weak solutions, subtra t the respe tive integral identities (1.62), and test it by the dieren e of these solution. This yields uniqueness. Cf. also Proposition 1.23(ii). The oer ivity (1.65b) an be ensured be various ways. One parti ular ase is
; ¡ 0; d £ 0; d ò L ( ) : a(x ; r; ) : % (x ; r; ) r £ p % d rq " b(x ; r)r £ b rq " b (x) ;b £ 0; b ò L ( ) :
d (x);
1
(1.68a)
1
(1.68b)
q £ 1 and d £ 0, b £ 0, min(d ; b ) ¡ 0. Then, £ p q q x y L p ;Rm,n % d y L q ;Rm % b y L q ;Rm " d L1 " b L1 . In parti ular, if d # 0, the oer ivity of the mapping A # A1 %A2 follows by Poin aré's inequality.
with some (
)
(
)
(
)
(
)
(
)
Remark 1.42 (NavierStokes system). One an apply Proposition 1.23 on a subspa e of W p ( ; Rm ). One example is for m # n and the subspa e
a Sobolev spa e
1;
Wdiv 0 ( ; Rn ) :# y ò W 1 p ( ; Rn ); div y # 0 on ; y  n # 0 1;
p
;
;
:
on
(1.69)
p # 2, a prominent example is the NavierStokes system for a velo ity y and a s alar variable (pressure) p des ribing a steady ow of an in ompressible vis ous (so alled For
Newtonian) uid:
%(y  x )y " y % x p # f
and
(x y)n t % byt # g
and
div y # 0 yn # 0
on
;
(1.70a)
on
;
(1.70b)
¡ 0 is the vis osity oe ient, % £ 0 mass density, and b # b(x) ¡ 0 is the sliding resistan e of the boundary wall, y # y n % y t is the de omposition of the velo ity on the boundary to the normal part y n :# ( y  n ) n and the tangential part y , and t analogously for the tra tion ve tor (x y ) n . We use the so alled Navier boundary on
where
dition, in luding nonpenetrability of the boundary wall. Let us note that the so alled
%(y  x )y # %div(y y) " %(div y)y # %div(y y) ts with the assumpn ¢ 3.116 In the weak formulation, the pressure disappears be ause, by Green's formula, P x p  y d x # P p ( y  n ) d S " P p div y d x # 0 for div y # 0 and
y  n # 0. The resulted weak formulation then onsists in the integral identity
onve tive term115
tion (1.64) if
: y ò Wdiv ( ; Rn ) : 1;2
115 for
;0
X x y : x y
% %(yx )y y dx % X byt  y t dS # X f  y dx :
More spe i ally, this onve tive term written omponentwise means [(
i
# 1; :::; n.
ÜÙ yy is ompa t from W ( ; Rn ) Ù L provided n ¢ 3 and 0 ¢ 2 " 4. Thus we pose a ( r; s ) # s " r r .
116
Note that the mapping
y
1;2
*
*
(2
y  x )y℄i
")/2 ( ;
(1.71)
# nk# y k xk y i
Rn
1
)
L ( ; Rn ) 2
1.4 Some dierential and integral equations
The oer ivity of the underlying operator on
X %( y
 x )
Ë 49
1 2 n Wdiv ( ; R ) is due to the al ulus 0 ;
;
y  y dx # X %(y y ) : x y dx
# X %(y  y )y  n dS " X %(yy) : x y % (y  y )div y dx
# "X %(y y) : x y dx # "X %( y  x )y  y dx
by Green's formula and by
(1.72)
div y # 0 in and by y  n # 0 on
, assuming
% ¡ 0 onstant.
Thus, all the integrals in (1.72) equal in fa t 0. For the mentioned oer ivity, (1.72) is
y # y. Let us note that the pointwise oer ivity (1.68a) does not hold, however. The uniqueness holds only for su iently small for e f and large vis osity oe ient , i.e. for smallturbulen e ows with with small (so alled) Reynolds' numbers. More 1 2 n spe i ally, it is natural to equipped W div 0 ( ; R ) with the norm used for
;
;
y
and then, assuming
1 %
y
2
dx % X byt dS
1/2
2
;
(1.73)
n ¢ 3, for two weak solutions y1 and y2 , we an estimate
# X (y
2
12 ;b
; b :# X x y
x )y1
1 
# X (y
" (y
x ) y 2  y 12
2 
x )y2
12 
# X (y
y12 % (y2 x )y12  y12 dx

x )y2
12 

y12 dx ¢ x y2 L2
¢ " b x y 2 ;
dx
;
2
;Rn,n ) y 12 L 4 ( ;Rn ) 2
(
L 2 ( ;Rn,n ) y 12 ; b 2
;
(1.74)
y12 :# y1 "y2 and b ¡ 0 is the onstant from the Poin arétype inequality b ( y L 4 ;Rn % y L 4 ;Rn ) ¢ y b . Therefore y12 # 0 prot vided x y 2 L 2 ;Rn,n 2 b /%. Testing (1.70a) and using (1.72), we obtain x y 2L2 ;Rn,n ¢ y 2 b # f  y L1 % g yt L1 ¢ f L4 3 ;Rn y L4 ;Rn % g L 4 3 ;Rn y L 4 ;Rn ¢ max( f L4 3 ;Rn ; g L4 3 ;Rn ) y b / b , from whi h we t obtain y b ¢ max( f L 4 3 ;Rn ; g L 4 3 ;Rn )/ b . Hen e, we obtain the unique
where
;
;
;
(
(
(
/
(
)
;
(
(
/
/
(
;
)
;
)
;
ness if
(
;
)
)
;
)
)
(
/
)
/
)
(
(
/
Remark 1.43 (Regularity).
(
;
)
/
(
;
)
;
)
max f L4 3 Rn ; g L4 3
(
/
)
Rn )
;
;
(
)
(
)
;
;
¢ b /% : 3
;
;
(1.75)
Sometimes, some additional qualitative information about
the solutions in addition to the basi quality
y ò W 1 p ( ; Rm ) is useful, in parti ular in ;
the ontext of optimal ontrol with state onstraints.
Ë
50
1.4.
1 Ba kground Generalities
Partial dierential equations of paraboli type
A further type of equations whi h we want to treat here as an example of an innitedimensional dynami al system is a system of Again we abbreviate
I :# [0; T℄
m
quasilinear paraboli equations.117
for a xed time horizon. More spe i ally, we will
onsider the Robintype (also alled NewtonFourier) initialboundaryvalue problem for a system of
m su h equations: y " div a(y; x y) % (y ; x y) # 0 t n  a(y; x y) % b(y) # 0 y(0; ) # y0
on
I, ;
I, ; on : on
/ 7 7
(1.76)
? 7 7 G
The basi natural requirement we will assume through the following text is that
a : , (Rm ,Rm,n ) Ù Rm,n ; b : (I , ) , Rm Ù Rm ; and
: (I , ) , (Rm ,Rm,n ) Ù Rm
/ 7
are Carathéodory mappings : ? 7 G
(1.77)
For notational simpli ity, in (1.76) and in what follows, we did and will not write expli itly the dependen e on
x
t.
and
The further natural requirement is a ontrolled
growth, namely
; ò L p ( ); C ò R :
; òL ; òL
# (p ") (
p" ")
(
(
I , ); C ò R :
I , ); C ò R :
where118
p" :#
np%2p n
a(x ; r; ) ¢
x % Crp
( )
b(t ; x ; r) ¢
(
t ; x ; r; ) ¢
(
p# :#
and
" /p "
t ; x) % Cr (
t ; x) % Cr
np%2p"2 n
% Cp" ;
p # "1"
1
;
p " "1"
provided
(1.78a)
and
(1.78b)
% C p¡
p/( p " " )
2 n %2 : n %2
;
(1.78 )
(1.79)
£ 0, the orresponding R , " Rm,n ) Ù L p (I , ; Rm,n ), p N : L (I , ; Rm ) , L p (I , ; Rm,n ) Ù
In parti ular, the growth onditions (1.78) ensures that, for Nemytski mappings work as
Na :
" L p (I , ;
m)
Nb : L (I , ; Rm ) Ù L (I , ; Rm ), and " L p (I , ; Rm ). Moreover, for ¡ 0, we an rely p#
p#
L p (I , ;
on respe tive ompa t embeddings
needed for existen e of weak solutions. Multiplying our equation by a test fun tion
y , applying the Green formula in spa e
together with using the Robintype boundary onditions, and making also bypart integration in time with using the initial ondition, we ome to the notion of the weak so
lution in the spirit of Remark 1.39: a fun tion
117
y ò L p (I; W 1 p ( ; Rm )) L ;
(
I; L2 ( ; Rm ))
For more details about su h equations we refer, e.g., to Gajewski et al. [342℄ or Lions [495℄ or also
[685℄.
118
L p (I; W 1;p ( )) : L p (I; W 1;p ( ))
The exponents in (1.79) are hosen in order to have the ontinuous embedding
L (I; L ( )) L (I; L2 ( ))
2
" L p (I
, ) and the ontinuous tra e operator u ÜÙ uI, # Ù L p (I , ), f. [685, Se t. 8.6℄. The ondition p ¡ (2n%2)/(n%2) is needed only for
optimizing the exponent
p# and an be avoided when (1.78b) would be strengthened.
Ë 51
1.4 Some dierential and integral equations
will be alled the weak solution to (1.76) if the following integral identity is fullled
T X X a ( y; x y ) : x y 0
% (y; x y)  y " y
y dx % X b(y)  y dS dt t
%X y(T)  y (T) dx # X y
for any
y ò W1
;
(
0
y (0) dx :

(1.80)
I , ; Rm ). Let us note that, supposing the growth onditions (1.78),
all the integrals in (1.80) are nite and the denition has a good sele tivity in the sense that, if the solution and the data are smooth enough, one an re over all three equation in (1.76) when, after making the bypart integration on
y
hoosing suitable test fun tions
I
and Green's formula on
,
0; T) , and then
rst with ompa t support on (
more general to re over the boundary and the initial ondition.119
Proposition 1.44 (Paraboli equations: existen e and uniqueness). y ò L ( ; Rm ), and semi oer ive
(1.78)(1.79) be satised,
X a ( y; x y ) : x y
Let
(1.65a)
and
2
0
% (t ; y; x y)  y dx % X b(t ; y)  y dS £ x y pLp
(
;Rm,n )
" ( t ) 1 % y L 2 Rm 2
;
(
)
(1.81)
ò L (I). Then the initialboundaryvalue problem (1.76) possesses just one p p ( ; R m )) L ( I ; L ( ; R m )) in the sense of (1.80) whi h adweak solution y ò L ( I ; W p (I; W p ( ; Rm ) ) % W m ditionally belongs also to W ( I ; L ( ; R )). If also a weak1
for some
1;
2
1;
1;
1;1
*
2
ened global monotoni ity
; ò L (I) : 1
t ò I :y; y ò W 1 p ( ; Rm ) : ;
a.a.
X a ( y; x y )
" a( y ; x y ) : x (y" y ) % (t ; y; x y) " (t ; y ; x y )  (y" y ) dx
% X b(t ; y) " b(t ; y )  (y" y ) dS £ " (t) y" y L2 Rm ; 2
;
(
(1.82)
)
holds, then this solution is unique.
V # # (P x  p dx) p , H # L ( ; Rm ), and p " " ( I , ; R m ) , L p # " ( I , ; R m ) with the duality the interpolation spa e Lp VH p # L ¢ 0, whi h proves (1.105b). Putting (1.105b) into (1.107), one an write the *
resulted inequality just in the form (1.105a).
is nitedimensional but int(D) # , we an work, instead of , with the linear hull of D , whi h is a losed subspa e of . Then D has nonempty interior with In ase
respe t to the relativized topology.
Convention 1.55. spa e
134
In fa t, the mappings and R need not be dened on the whole Z but only on the onvex subset K . Then the meaning of the dierential R(z) ò
This type of optimality onditions was rst invented by Fritz John (19101994) in [408℄. The asser
K has nonempty inte innitedimensional is admitted provided ertain additional assumptions on D and R are imposed. For ertain spe ial data , R , and D , an innitedimensional is also admitted in Ioe and Tikhomirov [399, Se t. 1.1.4℄. tion presented here is basi ally due to Casas [180, Thm. 5.2℄. For the ase that
rior, see also Zeidler [812, Se t. 48.3℄ where also
135
This proof is essentially due to Casas [180℄.
1.5 Basi s from optimization theory
Ë 63
L(Z ; ) of R at a point z ò K is that [R(z)℄( z "z) # lim"ÿ0 (R(z % "( z "z)) " R(z))/" z ò K only (and not for z ò Z as usual). The modi ation for is straightforward.
for
This may ause the dierentials to be determined uniquely only up to a losed linear subspa e provided
K is at. From the proof of Proposition 1.54, one an also see that R(z) : Z Ù su es to be dire tionally weakly ontinuous (i.e.
the linear operator
weakly ontinuous when restri ted on the segments), whi h is alled hemi ontinuity. Similarly for
(z) : Z Ù R. We will o
asionally use this onvention in what follows.
The rst multiplier
0
*
in the F. John onditions an sometimes degenerate to zero
falls
ompletely out.136 Therefore, the so alled normal ase 0 ¡ 0 (or equivalently 0 # 1) and then su h ondition be ome not mu h sele tive be ause the ost fun tion *
*
is of parti ular interests:
Proposition 1.56 (KarushKuhnTu ker onditions).137 Let K be onvex, and R be Gâteaux dierentiable, int( D ) #Ö , and z ò Argmin(P ). Let further one of the following onstraint quali ation hold:
: £ 0 ; #Ö 0 ; z ò K : *
*
[ R ( z )℄
*
*
; z "z 0 *
(1.108)
R is D onvex on K and ; z ò K : R( z ) 0 :
or
(1.109)
0 # 1.
Then (1.105) holds with
*
The ondition (1.109) is usually veriable quite simply, being alled the Slater
onstraint quali ation [728℄, while (1.108) is appli able to non onvex onstraint mappings
R, being alled the MangasarianFromowitz onstraint quali ation138 [515℄. 0 # 0 for a moment, (1.105 ) yields #Ö 0 £ 0. As £ 0 but #Ö 0, we simultaneously have # 0, f. (1.105b).
before, and where the equality is due to the orthogonality < Thus again we obtained
, a ontradi tion with (1.124). Here we used the fa t that, with 0 # ( z ) " ( z ), it holds149 Then
¢ 0. If < ; > # 0, then < ; N > would be a neighbourhood of 0, a ontradi tion.
149
Indeed,
*
*
150
*
*
*
*
*
The proof of (1.126) is analogous as those of (1.125).
*
Ë
72
Thus
1 Ba kground Generalities
S1 S2 # . Then we get (1.127a)(1.127 ) by the Eidelheit theorem as in the proof
of Proposition 1.54. The point (ii) follows as in Proposition 1.56 be ause the ontradi tion step uses
0 # 0 whi h then eliminates the ve torvalued from the onsiderations. *
An e ient straightforward approa h to multi riteria optimization is a so alled
s alarization, i.e. to onsider suitable s alarvalued riteria instead of the original ve torvalued one. In general, for a fun tional
F : 0 Ù
R, we an onsider the
s alarvalued problem Minimize subje t to
F (z) for z ò Z ; R(z) ¢ 0 ; z ò K
§
(1.129)
Inspired by the proof of Proposition 1.65, a worthy hoi e is a linear fun tional
F # *0
0 £ 0, 0 #Ö 0. Then any solution to (1.129) is D0 Slater optimal for (P ). If (Dad (P )) is onvex in 0 , then this linear s alarization overs even all D0 Slater op
with
*
*
*
timal solutions for (P ).151 Obviously, if
is D0  onvex, R is D onvex, and K is onvex, then also (Dad (P )) (Dad (P )),
is onvex and the linear s alarization is truly e ient. For a non onvex
F is parti ularly worth onsidering. For a nite number of s alarvalued n n % n fun tionals, i.e. if # ( i ) i #1 , 0 :# R , and D 0 :# (R ) , assuming i ¡ 0, it is more a nonlinear
e ient to take
F (0i )ni#1 # max (*0i 0i ) i #1 ; : : : ; n
with
0i £ 0 *
and
n * H 0i i #1
# 1:
(1.130)
D0 Slater optimal solution z, there is a suitable ntuple (0i )ni#1 for whi h z minimizes F , f. [614, Se t.2.1℄. *
Then, for any
1.5.d
Non ooperative game theory
In this se tion we mention briey some basi on epts and results from the theory of
non ooperative twoperson games.152 Having onned ourselves to two players (distinguished by the indi es 1 and 2),
Z l and two onvex sets K l Z l , as well as two ost fun tions l : Z 1 , Z 2 Ù R, l # 1 ; 2. The rst player uses the ontrol we will now onsider two lo ally onvex spa es
D0 Slater optimal z, put S # {(z) " ( z ); z ò Dad (P )}. Assuming S int(D0 ) #Ö z being D0 Slater optimal. Hen e, by the Eidelheit theorem, S * * * and int( D 0 ) an be separated by a linear fun tional, say , i.e. < ; int( D 0 )> ¡ 0 and < ; ( u ) " 0 0 0 * (Dad (P ))> ¢ 0. The former inequality gives 0 £ 0 while the latter one just says that z minimizes *0 over Dad (P ).
151
To show it, for any
, we would get a ontradi tion with
*
152
More about this topi an be found in the monographs by Aubin [35℄, Aubin and Ekeland [36℄,
Balakrishnan [49℄, or Zeidler [811, 812℄.
1.5 Basi s from optimization theory
Ë 73
z1 ò K1 with the aim to minimize the ost fun tion 1 , while the se ond player drives z2 ò K2 to minimize 2 . In ontext of game theory, the ontrols are also addressed as strategies.
1 # 2 , Z # Z1 , Z2 , K # K1 , K2 , and # 1 in the problem (P ) there. Hen e the a tual game begins if 1 #Ö 2 , i.e. if there Let us realize that if both players have identi al goals, whi h means
then we get basi ally the situation from Se t. 1.3 if put
is (to more or less extent) a oni t of goals. As one an anti ipate, game situations are also mathemati ally mu h more ompli ated than mere minimization problems, whi h orrespond to their ability to ree t in a more proper way the reality of live whi h is so dramati just due to frequently o
urring oni ting situations. For entirely non ooperative behaviour of two players, a suitable on ept of so
K1 ; K2 ; 1 ; 2 ) is the Nash equilibrium: z ; z2 ) ò K1 , K2 is alled a Nash equilibrium of the game (K1 ; K2 ; 1 ; 2 ) if
lution to the game des ribed by the data ( ( 1
1 (z1 ; z2 ) # min 1 ( z 1 ; z2 ) z 1 ò K 1
2 (z1 ; z2 ) # min 2 (z1 ; z 2 ) :
and
z2 ò K 2
(1.131)
Let us denote the set of all Nash equilibria by
Nash ( ; ) :# 1
K1 ,K2
2
(z1 ; z2 ) ò K1
, K ; (1:131) is satised :
(1.132)
2
Sometimes, Nash equilibria are also alled non ooperative equilibria, having the obvious meaning that ea h player follows only his or her individual prot and expe ts the same behaviour of the opponent. The existen e of the Nash equilibria often fails unless quite strong data quali ations are imposed; a tually it is not mu h surprising sin e everybody knows well from own everyday experien e that, willing to be in an equilibrium state, one should better avoid oni ting purely non ooperative situations. The following existen e theorem is, in fa t, equivalent with Brouwer's xedpoint Theorem 1.19 and is thus highly non onstru tive.153
Theorem 1.67 (Nash equilibria).154 Let the following assumptions be satised: are separately ontinuous ;
1 and 2 1 % 2
is jointly ontinuous on
(1.133a)
K1 , K2 ;
(1.133b)
:z ò K ; z ò K : (; z ) and (z ; ) are onvex ; ; ¡ 0 ;K onvex ompa t :z ò K : #Ö LevK1 (; z ) K 1
1
2
2
1
2
1;
153
2
2
(1.133 )
1
2
;
1
2
1;
;
(1.133d)
Inspe ting the proof of the Nash Theorem 1.67, we found even a series of non onstru tive argu
ments: a ontradi tion argument, a sele tion of nite overing relying on ompa tness, and the mentioned Brouwer xedpoint theorem.
154
John Nash, a 1994 Nobel prize winner for e onomy, formulated this theorem for a spe ial ase
where the set of admissible strategies
K1 and K2 are mixed strategies for a nite game, see [567℄.
74
Ë
1 Ba kground Generalities
; ¡ 0 ;K Then
2;
onvex ompa t
:z ò K : 1
1
#Ö LevK2 (z ; ) K 2
;
1
2;
:
(1.133e)
NashK1 ,K2 ( ; ) #Ö . 1
Proof. 155 If
K1
or
K2
2
K1 ,K2 and 1 ; 2 ) # NashK1 ,K2 (1 ; 2 ) thanks to the uniform oer ivity
is not ompa t, we an nd the Nash equilibrium on
realize that Nash K 1 , K 2 (
assumptions (1.133d,e). Let us abbreviate
u :# (u1 ; u2 ) ò K1 ,K2 and similarly z :# (z1 ; z2 ), and dene (z ; u) :# 1 (z1 ; u2 ) % 2 (u1 ; z2 ) :
We will show that minimum at
(1.134)
u $ (u1 ; u2 ) is a Nash equilibrium if and only if (; u) attains its
u, i.e.
:z ò K ,K : 1
(u ; u) ¢ (z ; u) :
2
(1.135)
1 (u1 ; u2 ) ¢ 1 (z1 ; u2 ) and 2 (u1 ; u2 ) ¢ 2 (u1 ; z2 ), f. (1.131). Conversely, if (1.135) holds, then for z1 :# u1 one gets
For the only if part, it su es to sum
1 (u1 ; u2 ) % 2 (u1 ; u2 ) # (u1 ; u2 ; u1 ; u2 )
¢ (u ; z ; u ; u ) # (u ; u ) % (u ; z 1
2
1
2
1
1
2
2 (u1 ; u2 ) ¢ 2 (u1 ; z2 ). Similarly, by putting z2 :# u2 1 (z1 ; u2 ).
so that
2
one gets
1
2)
1 (u1 ; u2 ) ¢
Suppose that there is no Nash equilibrium, whi h by (1.135) would mean that
:u ò K ,K ;z ò K ,K : 1
2
1
2
(u ; u) ¡ (z ; u) :
(1.136)
G z :# {u ò K1 ,K2 : (u ; u) ¡ (z ; u)}. By (1.133ab), all G z are open. Then G z }zòK1 ,K2 forms an open overing of K1 ,K2 . By ompa tness of K1 ,K2 there is a nite sub overing, i.e. there is {z i }i#1 n K1 ,K2 with some n ò N n su h that U i #1 G z i # K 1 , K 2 . This means pre isely
Denote
(1.136) just says that {
;:::;
:u ò K ,K ;j : (u ; u) ¡ (z j ; u) : 1
f i (u) :# max( (u ; u)" (z i ; u) ; 0). By (1.133ab), ea h f i f i £ 0 and, by (1.137), i f i ¡ 0. Furthermore, put
Put
'(u) :#
155
For general
(1.137)
2
n i H i (u)z i #1
and
i (u) :#
is ontinuous. Moreover
f i (u) : n j #1 f j ( u )
(1.138)
K1 and K2 onvex ompa t, this theorem has been proved by Nikaid and Isoda [577℄ nplayer generalization). It was further
by using the Brouwer xedpoint Theorem 1.19 (even for an
shown by Kindler [427℄ that, onversely, Brouwer's theorem follows from the NikaidIsoda theorem.
1.5 Basi s from optimization theory
As
'(K1 ,K2 ) o({z i }i#1
;:::;
n)
Ë 75
#: S, we have in parti ular '(S) S. As S is a ompa t
onvex nitedimensional subset, by Brower's xedpoint Theorem 1.19, there is some
u ò S su h that '(u) # u. Yet, by (1.137), (u ; u) ¡ (z j ; u) for a suitable j. By (1.133 ), (; u) is onvex. Thus be ause
due to
'(u) # u
by on
vexity n n # ('(u); u) # H i (u)z i ; u ¢ H i (u) (z i ; u) i #1 i #1 # j (u) (z j ; u) % H i (u) (z i ; u) (u ; u) i #Ö j # 0 if ¢ (u ; u) if
(u ; u)
(1.138)
f i (u) # 0
be ause
j (u) ¡ 0
f i (u) ¡ 0
j (u) (u ; u) % H i (u) (u ; u) # (u ; u) ; i#Ö j
whi h gives a ontradi tion.
Remark 1.68 (A spe ial ase: 2
1
and
2 ontinuous).
Let us still note that, if
1
and
themselves are jointly ontinuous, then the Nash theorem is an immediate onse
quen e of Kakutani's xedpoint Theorem 1.21 applied to the upper semi ontinuous
onvexvalued mapping
u#
K1 ,K2 Â± K1 ,K2 dened by
u1 Argmin1 (; u2 ) ÜÙ # Argmin ( ; u ) ; u2 Argmin2 (u1 ; )
the upper semi ontinuity follows essentially by Proposition 1.49. Cf. also Aubin and Ekeland [36, Se t. 6.3, Thm. 13℄ or [35, Thm. 12.2℄. In pra ti al omputer implementation, one is mostly for ed to approximate the
K2 by some (usually nitedimensional) sets K1d d d d and K 2 as well as the ost fun tions 1 and 2 by some 1 and 2 with d ¡ 0 being
set of admissible strategies
K1
and
an abstra t dis retisation parameter. Then immediately one asks whether the approximate problems onverge somehow to the original problem:
Proposition 1.69 (Convergen e of approximate games). (
K1d ; K2d ; 1d ; 2d )
satisfy (1.133a ) with
Let
K1d , K2d , 1d , 2d
(1.133)
be
satised,
K1 , K2 , 1 , 2 ,
in pla e of
respe tively, and let the following assumptions be satised:
:d £ d ¡ 0 : K d K d K ; K d onvex losed, lZ1
1
1
1
1
:d £ d ¡ 0 : K d K d K ; K d onvex losed, lZ2
2
:z ò K ; z ò K : 1
1
2
2
2
2
2
; ò R
d
%
(1.139a)
℄
K2d # K2 ;
d ¡0
(1.139b)
C
1
C
1d (z1 ; ) Ù 1 (z1 ; ) & 2d (; z2 ) Ù 2 (; z2 );
(1.139 )
C
1 % 2 Ù 1 % 2 ; d
#K ;
d ℄ K1 d ¡0
;K
1;
onvex ompa t
(1.139d)
:d ò R :z ò K : LevK1d inf 1d K1d z2 % d (; z ) K % ;
d
2
(
2
;
)
1
2
1;
;
(1.139e)
76
Ë
1 Ba kground Generalities
; ò R% ;K
2;
:d ò R% :z ò K d : LevK2d inf 2d z1 K2d % d (z ; ) K
onvex ompa t
1
;
Then, for all
1
(
;
1
2
)
2;
:
(1.139f)
d ¡ 0, NashK d ,K d (1d ; 2d ) #Ö and 1
2
Limsup Nash ( d ; d ) Nash ( ; ) : 1
K 1d , K 2d
d Ù0
2
1
K1 ,K2
(1.140)
2
Proof. First, let us note that, by Theorem 1.67, the approximate problems always ad
K1d K1 and K2d K2 are onvex ompa t. Hav
mit Nash equilibria; note that both
;
;
z1d ; z2d ) ò NashK1d ,K2d (1d ; 2d ), by the uniform oer ivity of approximate problems (1.139e) and (1.139f) one an lo alize all onsiderations on a ompa t set K 1 , K 2 and suppose that (possibly after taking a ner net) there is ( z 1 ; z 2 ) ò K 1 , K 2 su h that z1d Ù z1 and z2d Ù z2 for d Ù 0. Our aim is to show that (z1 ; z2 ) òNash K1 ,K2 (1 ; 2 ). d d d d d d d d d d We know that 1 ( z 1 ; z 2 ) ¢ 1 ( z 1 ; z 2 ) and 2 ( z 1 ; z 2 ) ¢ 2 ( z 1 ; z 2 ) for any d d ( z 1 ; z 2 ) ò K , K . In parti ular, 1 2 ing (
;
;
:( z ; z ) ò K d , K d : d (z d ; z d ) % d (z d ; z d ) ¢ d ( z ; z d ) % d (z d ; z ) : 1
2
1
2
1
1
2
2
1
2
1
1
2
2
(1.141)
2
1
limdÙ d ( z ; z d ) # ( z ; z ) and limdÙ d (z d ; z ) # (z ; z ). Mored d d d d d over, by (1.139d) also lim d Ù ( z ; z ) % ( z ; z ) # ( z ; z ) % ( z ; z ). This By (1.139 ),
0
1
1
1
2
0
1
1
2
1
0
2
2
1
2
2
1
1
2
2
1
2
1
2
1
2
2
allows us to pass to the limit in (1.141), whi h gives
:( z ; z ) ò K d , K d : (z ; z ) % (z ; z ) ¢ ( z ; z ) % (z ; z ) : 1
2
1
1
2
1
2
2
1
2
1
1
2
2
1
2
(1.142)
Eventually, by (1.133a) with (1.139a) and (1.139b) one an see that (1.142) holds even for any (
z 1 ; z 2 ) ò K1 ,K2 . In parti ular, taking z 1 :# z1 shows that 2 (z1 ; z2 ) ¢ 2 (z1 ; z 2 ) z 2 ò K2 . Analogously, z 2 :# z2 shows that z1 minimizes 1 (; z2 ) over K1 .
for any
In view of the (quite restri tive) onditions (1.133b) and (1.139d), it is worth
1 % 2 is onstant without any loss of 1 % 2 # 0. This means that the players have entirely
onsidering a spe ial lass of games where generality, we an suppose
antagonisti goals in the sense that the prot of one player is just the loss of the other one and vi e versa. In su h situation we speak about a zerosum game. Putting
:# 1 # "2 , from (1.131)
one an easily see that the point (
z1 ; z2 ) ò K1 ,K2
is a
Nash equilibrium if and only if
min ( z ; z
z 1 ò K 1
1
2)
# (z ; z 1
Su h point is also alled a saddle point of
2)
# max (z ; z ) : 1
z2 ò K 2
, and
2
(1.143)
is addressed as a payo. Let us
denote the set of all saddle points by
Saddle :# Nash (; ") # K1 ,K2
K1 ,K2
( z 1 ; z 2 ) ò K 1 , K 2 ;
1:143) holds :
(
(1.144)
Ë 77
1.5 Basi s from optimization theory
The fa t that (
z1 ; z2 ) ò K1 ,K2 is a saddle point of is equivalent156 to the fa t that
inf sup ( z ; z 1
z 1 ò K 1 z 2 ò K 2 and
# sup inf ( z ; z
2)
1
z2 ò K 2 z1 ò K 1
(1.145)
2)
z1 ò K1 and z2 ò K2 are so alled onservative strategies in the sense that
sup (z ; z ) # inf sup ( z ; z 1
z 2 ò K 2
2
1
z 1 ò K 1 z 2 ò K 2
2)
inf ( z ; z ) # sup inf ( z ; z ) :
and
1
z 1 ò K 1
2
1
z 2 ò K 2 z1 ò K 1
2
The meaning of a onservative strategy is that a player tries to rea h the highest own prot on the assumption that the only goal of the opponent is to make him or her as highest harm as possible.157 As a plain onsequen e of Theorem 1.67, we an laim that provided
is separately ontinuous and
(; z2 )
has a saddle point
is onvex and inf ompa t while
(z1 ; ) is on ave and sup ompa t and uniformly oer ive, and K1
and
K2
are on
vex. Nevertheless, spe ial hara ter of the zerosum problem makes possible to modify the oer ivity assumptions:
Theorem 1.70 (Saddle point von Neumann [781℄, generalized).158 Let
is separately ontinuous ;
(1.146a)
K1 and K2 are onvex;
:z ò K ; z ò K : ;z ò K : ò R : ;z ò K : ò R : Then
1
1
2
2
1
1
2
(1.146b)
(; z2 ) is onvex; (z1 ; ) is on ave,
2
(1.146 )
LevK1 (; z ) is ompa t; LevK2 ("(z ; )) is ompa t :
(1.146d)
2
;
(1.146e)
1
;
SaddleK1 ,K2 #Ö .
Sket h of the proof. We use Theorem 1.67 for
K in # {z i ò K i ;
z i ¢ n}.
Thus, for a
n, a saddle point of or, in other words, a Nash equilibrium (u1n ; u2n ) ò Nash K n , K n ( ; " ) does exist. For z i # z i , we then have 1 2 su iently large
for
z2
"
# z
for
2
z1
inf (; z ) ¢ (u n ; z ) ¢ (u n ; u n ) ¢ (z ; u n ) ¢ sup (z ; ) 2
1
2
1
2
1
2
# z
1
1
% :
(1.147)
156
See, e.g., Aubin [35, Proposition 8.1℄ or Aubin and Ekeland [36, Se t. 6.2, Proposition 1℄.
157
If no onvex/ on ave stru ture of the game an be guaranteed (as typi al, e.g. in games with fully
nonlinear systems or pursuer/evader games), it is often a satisfa tory task to nd a onservative strategy of at least one of the players; f. Friedman [334℄, M Millan and Triggiani [525℄, Nikol'ski [579℄, Warga [791, Chap. IX℄, et .
158
See [781℄ for a spe ial ase that the set of admissible strategies
a nite game, or Nikaid and Isoda [577℄ for
K1
and
K2
K1 and K2 are mixed strategies for
onvex and ompa t. The presented general
version is due to Aubin and Ekeland [36, Se t. 6.2, Thm. 8℄ where even a lower/upper semi ontinuous payo fun tion
is admitted.
78
Ë
1 Ba kground Generalities
u1n }nòN and {u2n }nòN must be bounded, hen e they live in some ompa t set K 1 , K 2 for m large enough, and thus (up to possibly a subn n m m n n sequen es) ( u 1 ; u 2 ) Ù ( u 1 ; u 2 ) ò K 1 , K 2 and also ( u 1 ; u 2 ) onverges to some limit, say L . Making a limit passage in (1.147) gives
This implies that the sequen es {
m
m
(u1 ; z2 ) ¢ lim inf (u1n ; z2 ) ¢ lim (u1n ; u2n ) # L ¢ lim sup (z1 ; u2n ) ¢ (z1 ; u2 ) : n Ù
n Ù
n Ù
(1.148) Putting
z1 # u1 and z2 # u2 , we get (u1 ; u2 ) # L. Then (1.148) yields (u1 ; u2 ) that a on K1 ,K2 .
saddle point of
1 (; z2 ) and 2 (z1 ; ) possess Gâteaux derivatives, denoted respe tively by and z 2 2 , from (1.131) we an easily establish the rstorder ne essary onditions for the Nash equilibrium point ( z 1 ; z 2 ), namely If
z
1 1
z
1
1 (z1 ; z2 ) ò "N K1 (z1 )
and
z2
2 (z1 ; z2 ) ò "N K2 (z2 ) :
Conversely, (1.133 ), (1.146b) and (1.149) imply (
(1.149)
z1 ; z2 ) ò NashK1 ,K2 (1 ; 2 );
f. Re
mark 1.58. Let us now investigate a gametheoreti al problem involving a state equation like in Se tion 1.2d, i.e.
(P )
where
J1 (z1 ; z2 ; y) ; . Nash equilibrium 6 6 J2 (z1 ; z2 ; y) ; 6
(z1 ; z2 ; y) # 0 ; z1 ò K1 ; z2 ò K2 ;
> subje t to 6 6 6 F
J l : Z1 , Z2 , Y Ù
R, l # 1; 2, and : Z , Z , Y Ù X with Y and X Bana h 1
2
spa es. Like in Se tion 1.1.2d, we suppose that the state equation always a unique solution
y # (z1 ; z2 )
(z1 ; z2 ; y) # 0 has
whi h denes the ontroltostate mapping
: K1 ,K2 Ù Y . Then we dene naturally the set of equilibrium points of (P ) by
Nash(P ) :# Nash( ; K1 ,K2
1
2)
for
l (z1 ; z2 ) :# J l (z1 ; z2 ; (z1 ; z2 )); l # 1; 2:
Theorem 1.67 and Proposition 1.69 an be applied straightforwardly to (P ); note that
l whi h is biane159.
the assumption (1.133 ) about the onvex stru ture of the omposed ost fun tions basi ally for es us to onsider only the ontroltostate mapping
It is noteworthy to spe ify the optimality onditions (1.149) for this problem involving the state equation:
Proposition 1.71 (Optimality onditions for (P )). Let J l (z ; z ; ) : Y Ù R, l # 1; 2, and (z ; z ; ) : Y Ù X be Fré het dierentiable, J (; z ; y) : Z Ù R, J (z ; ; y) : Z Ù 1
1
2
1
2
2
1
2
1
2
This means both ( ; z 2 ) and ( z 1 ; ) are ane. In fa t, the uniform onvexity of J 1 ( ; z 2 ; y ) and J2 (z1 ; ; y) may sometimes guarantee (1.133 ) even if (; u2 ) and (u1 ; ) are slightly nonane, f.
159
[627, 679℄.
Ë 79
1.5 Basi s from optimization theory
R,
( ; z 2 ; y ) : Z 1 Ù X , and ( z 1 ;  ; y ) : Z 2 Ù X be Gâteaux equidierentiable around y ò Y , let the ontroltostate mapping : K1 ,K2 Ù Y as well as all the mappings [ z 1 J 1 ( z 1 ; z 2 ; )℄( z 1 ) : Y Ù R, [ z 2 J 2 ( z 1 ; z 2 ; )℄( z 2 ) : Y Ù R, [ z 1 ( z 1 ; z 2 ; )℄( z 1 ) : Y Ù X , and [z2 (z1 ; z2 ; )℄( z 2 ) : Y Ù X be ontinuous, let y (z1 ; z2 ; y) ò L(Y; X )
have a bounded inverse, and (1.146b) be valid. Then: (i)
If ( z 1 ;
z2 ) òNash(P ) and y # (z1 ; z2 ), then [ z l
for
1 ; 2 ò X *
*
(z1 ; z2 ; y)℄ l " z l J l (z1 ; z2 ; y) ò N K l (z l ) ;
*
*
*
l # 1; 2;
(1.150)
satisfying the adjoint equation [ y
(z1 ; z2 ; y)℄ l # *
*
y
J l (z1 ; z2 ; y); l # 1; 2:
(1.151)
if, for some ( z 1 ; z 2 ) ò K1 ,K2 , the omposed ost fun tions J1 (; z2 ; (; z2 )) : K1 Ù R and J2 (z1 ; ; (z1 ; )) : K2 Ù R are onvex and (1.150) (1.151) hold for y # ( z 1 ; z 2 ) and for 1 ; 2 ò X , then ( z 1 ; z 2 ) òNash(P ).
(ii) Conversely,
*
*
*
Sket h of the proof. Sket h of the proof. The point (i) is just (1.149) if one evaluates z 1 1 ( z 1 ; z2
z2 ) and 2 (z1 ; z2 ) by means of Lemma 1.59. The su ien y (i.e. the point (ii)) then follows
by the onvex stru ture of the parti ular minimization problems; f. Remark 1.58. For a spe ial ase
J # J1 # "J2 , (P ) be omes the zerosum game problem involving
a state equation:
J(z1 ; z2 ; y) ;
Minimax . 6 6
(P )
(z1 ; z2 ; y) # 0 ; z1 ò K1 ; z2 ò K2 ;
subje t to > 6 6 F
0
and it is natural to dene the set of saddle points of (P0 ) by
Saddle(P ) :# Saddle 0
for
K1 ,K2
(z1 ; z2 ) :# J(z1 ; z2 ; (z1 ; z2 )) :
Corollary 1.72 (Optimality onditions for (P )). Let J(z ; z ; ) : Y Ù R be Fré het difz ; y) : Z Ù R and J(z ; ; y) : Z Ù R be Gâteaux equidierentiable around y ò Y , let the mappings [ z 1 J ( z ; z ; )℄( z ) : Y Ù R and [ z 2 J ( z ; z ; )℄( z ) : Y Ù R be ontinuous, let (1.146b) be valid, and let and be as in Proposition 1.71.
ferentiable, J ( ;
1
0
2
1
1
2
2
1
2
1
1
2
2
Then: (i)
If ( z 1 ;
z2 ) òSaddle(P0 ) and y # (z1 ; z2 ), then J z ; z2 ; y) " [z1 (z1 ; z2 ; y)℄ ò "N K1 (z1 ) ;
(1.152a)
J z1 ; z2 ; y) " [z2 (z1 ; z2 ; y)℄ ò N K2 (z2 ) ;
(1.152b)
*
z1 ( 1 z
with
òX *
*
*
2 (
*
*
satisfying the adjoint equation [ y
(z1 ; z2 ; y)℄ # *
*
J z ; z2 ; y) :
y ( 1
(1.152 )
80
Ë
1 Ba kground Generalities
z1 ; z2 ) ò K1 ,K2 , J(; z2 ; (; z2 )) : K1 Ù R is onvex, J(z1 ; ; (z1 ; )) : K2 Ù R is on ave, and (1.152) hold for y # (z1 ; z2 ) and for ò X , then (z1 ; z2 ) òSaddle(P0 ).
(ii) Conversely, if, for some (
*
*
2 Theory of Convex Compa ti ations Ar himedes denes a onvex ar ... When in the seventeenth
entury
Ar himedes'
methods
were
taken up again, onvexity ... played still a role, for instan e in the work of Fermat.
Moritz Werner Fen hel ...though
onvex
sets
belong
to
(19051988)
geometry,
they
be ome one of the basi tools of the analyst...
Gustave Choquet Aleksandrov
began
to
onstru t
(19152006)
the
theory
of
ompa t spa es... This on ept ... still today is used
onstantly in various elds of mathemati s.
Evgeniy Frolovi h Mis henko
(19222010)
In various relaxation s hemes the ommon feature is the onvexity of the used
ompa t envelopes of the original spa es. Thus, to give an abstra t and unied viewpoint to parti ular on rete ases, it is worth developing a general theory of what we will all onvex ompa ti ations. This is, as it sounds, ompa ti ations whi h are simultaneously onvex subsets of some lo ally onvex spa es. The onvexity is
ertainly a onsiderable restri tion and it should be emphasized that not every topologi al spa e admits nontrivial onvex ompa ti ations but, on the other hand, there are topologi al spa es whi h admits a lot of them. It is then ertainly useful to introdu e a natural ordering of onvex ompa ti ations of a given spa e. This will be done in Se tion 2.1. Furthermore, we will nd useful to have a ertain unied (we will say anoni al) form of an arbitrary onvex ompa ti ation. Imitating the lassi al onstru tion based on the multipli ative means on some ring of ontinuous bounded fun tions, in Se tion 2.2 we will onstru t our onvex ompa ti ations by using the means ( f. Se tion 1.5) on a suitable (we will say onvexifying) linear subspa e of the spa e of ontinuous bounded fun tions on a topologi al spa e to be ompa tied. An important result is that there is a onetoone orderpreserving orresponden e between all losed onvexifying subspa es and all onvex ompa ti ations. In parti ular, it identies the topology of the uniform onvergen e as de isive for the reated onvex
ompa ti ation in the sense that, on one hand, making a losure of the onvexifying linear subspa e in this topology does not hange the orresponding onvex ompa ti ation but, on the other hand, any further enlargements reate onvex ompa ti ations whi h are a tually stri tly ner. In many of on rete applied problems the spa es to be ompa tied possess, beside a topologi al stru ture, also a bornology al stru ture. It enables us to speak about
https://doi.org/10.1515/9783110590852002
Ë
82
2 Theory of Convex Compa ti ations
a oer ivity of these problems, whi h eventually lo alizes investigations onto one suf iently large bounded set. Typi ally this set annot be hosen a priori for a given
lass of problems, whi h for es us to modify in Se tion 2.3 our on ept of onvex om
 ompa t but,  ompa ti ation. It
pa ti ations in su h a manner that the resulting envelope is onvex in general, not ompa t. As su h, it will be alled a onvex
may be itself a linear manifold, though typi ally it is rather not. Sometimes onvex
 ompa ti ations an have additional important pa tness, metrizability, or so alled
spe ial properties, as lo al om
B  oer ivity.
The anoni al form enables us, in Se tion 2.4, to develop an approximation theory of onvex ompa ti ations, whi h forms an abstra t framework for developing a omputerimplementable numeri al s hemes in on rete ases. Also, the anoni al form enables us to formulate simple riteria for mappings (esp. fun tions) to admit an ane ontinuous extensions onto respe tive onvex
 ompa ti ations and also to
investigate their dierentiablity properties. This will be performed in Se tion 2.5.
Convex ompa ti ations
2.1
Let us begin dire tly with the denition of the notion of a onvex ompa ti ation whi h represents the fundamental on ept used for relaxation theory as presented in this book. Let us onsider a topologi al spa e
U
to be ompa tied,
T
being its
topology.
Denition 2.1 (Convex ompa ti ation).
A triple ( K ;
Z ; i) is alled a onvex ompa t
i ation of a topologi al spa e ( U; T ) if (a)
Z is a Hausdor lo ally onvex spa e,
(b)
K is a onvex, ompa t subset of Z ,
( )
i : U Ù K is ontinuous, and
(d)
i(U) is dense in K .
If
i
is also inje tive (resp. homeomorphi al embedding), ( K ;
(resp.
Z ; i)
is alled a Hausdor
T  onsistent) onvex ompa ti ation.
K ; i), reated from a onvex ompa ti ation (K ; Z ; i) by Z , is a ompa ti ation of U in a usual sense as introdu ed in Se tion 1.1. Also note that, in general, the embedding i is even not required to be inje tive so that some points of the original spa e U an be glued together in the ompa ti ation K . The set of all onvex ompa ti ations of a given topologi al spa e U an be natLet us note that the pair (
forgetting
urally ordered.
Denition 2.2 (Ordering of onvex ompa ti ations).
and ( K 2 ;
Let
us
onsider
Z2 ; i2 ) two onvex ompa ti ations of U . Then we will say that:
(
K1 ; Z1 ; i1 )
2.1 Convex ompa ti ations
(i)
Ë 83
K1 ; Z1 ; i1 ) is a ner onvex ompa ti ation of U than (K2 ; Z2 ; i2 ), and write K1 ; Z1 ; i1 ) ³ (K2 ; Z2 ; i2 ), if there is an ane ontinuous mapping : K1 Ù K2 1 1 1 1 xing U ; the adje tive ane means ( z % z ) # 2 2 2 (z) % 2 ( z ) for any z ; z ò K1 , while xing U means i1 # i2 . ( K 1 ; Z 1 ; i 1 ) and ( K 2 ; Z 2 ; i 2 ) are equivalent with ea h other, and write ( K 1 ; Z 1 ; i 1 ) Ê ( K 2 ; Z 2 ; i 2 ), if simultaneously ( K 1 ; Z 1 ; i 1 ) ³ (K2 ; Z2 ; i2 ) and (K2 ; Z2 ; i2 ) ³ ( K 1 ; Z 1 ; i 1 ). ( K 1 ; Z 1 ; i 1 ) is stri tly ner than ( K 2 ; Z 2 ; i 2 ), and write ( K 1 ; Z 1 ; i 1 ) ± ( K 2 ; Z 2 ; i 2 ), if ( K 1 ; Z 1 ; i 1 ) ³ ( K 2 ; Z 2 ; i 2 ) and ( K 1 ; Z 1 ; i 1 ) ÊÖ ( K 2 ; Z 2 ; i 2 ). (
(
(ii)
(iii)
K1 ; Z1 ; i1 ) ³ (K2 ; Z2 ; i2 ), we will also say that (K2 ; Z2 ; i2 ) is Z1 ; i1 ), and write (K2 ; Z2 ; i2 ) ² (K1 ; Z1 ; i1 ). Let us emphasize that ( K 1 ; Z 1 ; i 1 ) Ê ( K 2 ; Z 2 ; i 2 ) does not mean that the lo ally onvex spa es Z 1 and Z 2 are isomorphi with ea h other. Also let us note that, if Z 's are forgotten, the orderOf ourse, if (
oarser than ( K 1 ;
ing of onvex ompa ti ations agrees with the usual ordering of ompa ti ations
K1 ; Z1 ; i1 ) ³ (K2 ; Z2 ; i2 ), then the ompa ti ation ( K 1 ; i 1 ) is ner in the usual sense than ( K 2 ; i 2 ). Of ourse, the onverse as introdu ed in Se tion 1.1. For example, if (
impli ation does not hold in general. In parti ular, the ane ontinuous mapping
:K ÙK 1
2
xing
U , used in Def
inition 2.2, must be always surje tive be ause this holds for usual ompa ti ations, as well.1
(
Let us also agree that we will o
asionally abbreviate e.g. K 1 ³ K 2 instead of K1 ; Z1 ; i1 ) ³ (K2 ; Z2 ; i2 ) when Z1 , Z2 , i1 , and i2 are obvious from a ontext. The set of all onvex ompa ti ations of a given topologi al spa e U has always
the smallest element, i.e. the oarsest onvex ompa ti ation. Indeed, any triple
K # {z} a singleton in a Hausdor lo ally onvex spa e Z and with i z) is the oarsest onvex ompa ti ation of U . Obviously, this
onvex ompa ti ation glues all points of the original spa e U into one point and thus the embedding i : U Ù K # { z } is not inje tive provided U ontains at least two points. Sometimes it an happen that U does not admit any other (up to the equiva
(
K ; Z ; i)
with
onstant (and equal
len e) onvex ompa ti ation than this ollapsed oarsest one. For example, this happens when
U
is a dis rete topologi al spa e ontaining only a nite number of
points. Fortunately, in pra ti ally interesting ases the set of all onvex ompa ti ations an be far ri her, f. Theorem 3.41 below. Ex eptionally it an happen that
U possesses also the nest onvex ompa ti aU . However, generally the nest
tion, e.g. in the previous ase of a nite dis rete spa e
onvex ompa ti ation does not exist and, instead of it, we have only guaranteed an
1
z ò K there is a net {u } ò U su h that i (u ) Ù z in Z . As K K must have a luster point z ò K . As : K Ù K is ontinuous, is surje tive sin e z ò K was arbitrary.
Indeed, by Denition 2.1d for any
i u )} ò
is ompa t, the net { 1 (
z
( )
# z and
2
1
2
1
2
1
2
2
1
84
Ë
2 Theory of Convex Compa ti ations
existen e of at least one maximal onvex ompa ti ation.2 This is an essential dieren e between the ordering of the usual ompa ti ations. We already know that, if
U
is a regular topologi al spa e, this ordering always admits a nest ompa ti ation, namely the e hStone ompa ti ation
U . The e hStone ompa ti ation an be
in simple ases homeomorphi with a maximal onvex ompa ti ation but in general it need not be homeomorphi with any onvex ompa ti ation; e.g. when
U
is
nite dis rete, ontaining at least two points.
Example 2.3 (Maximal onvex ompa ti ations).
Let U be homeomorphi al via i K of a lo ally onvex spa e Z . Then (K ; Z ; i) is obviously a Hausdor onvex ompa ti ation of U . Besides, it is maximal.3 On the other hand, su h K need not be the nest onvex ompa ti ation. This appears in the simple ase 2 indi ated on Figure 2.1 where U R is homeomorphi al both to an ellipse K 1 and to a re tangle K 2 . As we just saw, ea h of them forms a maximal onvex ompa ti ation of U whi h, however, annot be transformed anely one onto ea h other. Therefore,
with a onvex ompa t subset
these maximal onvex ompa ti ations are not equivalent with ea h other and, in parti ular, there does not exist any nest onvex ompa ti ation.
K2
K1
i1
PSfrag repla ements
U
i2 Fig. 2.1:
Two dierent maximal
onvex ompa ti ations.
2.2
Canoni al form of onvex ompa ti ations
The aim of this se tion is to show that every onvex ompa ti ation of alently expressed as a set of means M(F ) on a suitable subspa e of all ontinuous, bounded, realvalued fun tions on
f
norm by C 0 ( U )
2
:# supuòU f(u). Supposing that F
F
U an be equivC0 (U)
of the ring
U endowed with the Chebyshev
ontains onstants, we have de
See Corollary 2.9 and realize that there always exists at least one onvex ompa ti ation, namely
the oarsest one.
K1 ; Z1 ; i1 ) were a onvex ompa ti ation of U ner than K and were the ane onK1 Ù K xing U , then "1 # i1 i"1 would be ontinuous, whi h shows that K1 is also a oarser onvex ompa ti ation than K . This shows K to be maximal. 3
Indeed, if (
tinuous surje tion
2.2 Canoni al form of onvex ompa ti ations
ned in Se tion 1.2. the set of all means on
F ) :# ò F * ; F
M(
# { : F F*
where
# 1 and ; 1 # 1 Ù R linear; £ 0 & ; 1 # 1} ;
denotes the topologi al dual of
f ò F , by
by4
*
(1.21). Moreover, the evaluation mapping and
F
F
e(u); f
U . For this, the following property of F
(2.1)
endowed with the standard dual norm
e :U Ù
M(
F ) has been dened, for u ò U
:# f(u) :
It should be emphasized that, in general, (M(F )
(2.2)
; e) need not be a ompa ti ation of
is de isive, as we an see in Proposition 2.5.
Denition 2.4 (Convexifying subspa es). ing if
Ë 85
C (U) is alled onvexify0
F
A subspa e
1 1 :u ; u ò U ; a net {u } ò U :f ò F : lim f(u ) # f(u ) % f(u ): 2 2 ò 1
1
2
(2.3)
2
Let us note that any subspa e of a onvexifying subspa e is again onvexifying but not vi e versa. Thus requiring a subspa e
ertain impli it restri tion that
F
F
C (U) to be onvexifying represents a 0
must not be too large.
Proposition 2.5 (Properties of M(F )).
be a linear subspa e of
C0 (U) ontaining
onstant fun tions and M(F ) be endowed with the weak* topology of
F * . Then the fol
Let
F
lowing statements are equivalent with ea h other: (i)
F
(ii)
e(U) is dense in M(F ),
is onvexifying,
(iii) the pair (M(F )
; e) is a ompa ti ation of U ,
(iv) the triple (M(F )
; F ; e) is a onvex ompa ti ation of U . *
e is inje tive if and only if F u1 #Ö u2 ;f ò F : f(u1 ) #Ö f(u2 ).
Besides,
separates points of
U in the sense that :u1 ; u2 ò U
ã(iii)ã(iv) is obvious.
Proof. In view of Proposition 1.16, the equivalen e (ii)
á(ii). Let us put
We will prove (i)
F :#
{nite subsets of
F} , N
(2.4)
£ dened by # (F ; n ) £ # (F ; n ) i F F and n £ n . Clearly, F is dire ted by £. From (2.3) we an dedu e that for every f ò F 1 1 and n ò N there is f n ò su h that f ( u ) " 2 f(u ) " 2 f(u ) ¢ n for all £ f n . As ( ; £) is a dire ted index set, for every # ( F; n ) ò F there is ò su h that £ f n for all f ò F . We denote ({ f } ; n ) ò F by f n . Putting u # u , we get ! ! 1 1 1 :f ò F ; n ò N; £ f n : !!!!! f( u ) " f(u ) " f(u ) !!!!! ¢ : 2 2 n and endow it with the ordering 1
1
1
1
2
2
2
1
2
2
1
;
1
2
;
;
;
;
4
For the equality in (2.1), see Proposition 1.16.
1
2
Ë
86
2 Theory of Convex Compa ti ations
u }òF
for
will play then the role of { } ò .
Now, let us onsider another point we get, for ea h
F
in Denition 2.4 without any u u3 ò U . Repla ing u1 and u2 with u3 and u ,
In parti ular, we an take the ommon index set loss of generality of ourse, {
ò F , a net {u } òF
su h that
! ! 1 1 1 :f ò F ; n ò N; £ f n : !!!!! f(u ) " f(u ) " f( u ) !!!!! ¢ : 2 2 n 3
;
u
By the diagonalization pro edure, we obtain the net { } ò F su h that
! ! 1 1 1 :f ò F ; n ò N; £ f n : !!!!f(u ) " f(u ) " f(u ) " f(u ) !!!! ¢ ! ! 4 4 2 !! !! 1 1 1 !!!! 1 1 !! !! ¢ !!!f(u ) " f( u ) " f(u )!! % !!f( u ) " f(u ) " f(u ! ! 2 2 2 2 2 ! ! ! 1
;
2
3
3
1
!! !! !! !!
2)
¢
3 : 2n
fu
u
In other words, we have got the net { } ò F su h that { ( )} ò F onverges to
1 1 1 4 f(u1 ) % 4 f(u2 ) % 2 f(u3 ) for all f ò F . It is now evident that repeating this pro edure k
yields su h a net for every onvex ombination j #1
"l {m2 ; l ò
N; m #
0; :::; 2l }, and kj#1
a j f(u j ) with k ò N; u j ò U , a j ò L #
a j # 1. M L (F ) the set of all nite means ò M(F ) in the form # k j #1 a j e ( u j ) with a j ò L . Paraphrasing the pre eding on lusion, we an also say that for every ò M L (F ) there is a net, say { u } ò F , su h that e ( u ) Ù weakly*. In other words, e ( U ) is dense in M L (F ). n Sin e L is dense in the interval [0 ; 1℄ and the mapping ( a 1 ; :::; a n ) ÜÙ j #1 a j e ( u j ) n n from {( a 1 ; :::; a n ) ò [0 ; 1℄ ; j #1 a j # 1} to M(F ) is ontinuous, M L (F ) is dense in the Let us denote by
set of all nite means whi h is dense in M(F ) by Proposition 1.16. It eventually yields that
e(U) is dense in M(F ).
á(i). Sin e M(F ) is 1 1
onvex, # 2 e(u1 ) % 2 e(u2 ) belongs to M(F ) for ea h u1 ; u2 ò U . As e(U) is supposed to be dense in M(F ), there is a net { u } ò U su h that { e ( u )} ò onverges to . It 1 1 means pre isely that lim ò e ( u )( f ) # [ e ( u 1 ) % 2 2 e(u2 )℄(f) for all f ò F , whi h is just 1 1 1 1 (2.3) be ause e ( u )( f ) # f ( u ) and [ e ( u 1 ) % 2 2 e(u2 )℄(f) # 2 f(u1 ) % 2 f(u2 ). Finally, if u 1 #Ö u 2 and F separates u 1 ; u 2 ò U , then there is f ò F su h that It remains to prove the onverse impli ation, that means (ii)
e(u1 ); f> # f(u1 ) #Ö f(u2 ) # , this means e(u1 ) #Ö e(u2 ), thus e is inje tive. e is inje tive, then u1 #Ö u2 implies e(u1 ) #Ö e(u2 ), whi h means the existen e of f ò F su h that f ( u 1 ) # < e ( u 1 ) ; f > #Ö < e ( u 2 ) ; f > # f ( u 2 ), hen e F separates u1 and u2 whi h were taken arbitrarily. Å
# f (z); note that, be ause i(U) is dense in K , there is pre isely one f ò A (K) su h that f # f i, hen e is well dened. As all f are ontinuous and ane, so is . Be ause
where
Let us dene a mapping
for any
uòU
and
i u)); f # f (i(u)) # f(u) # e(u); f
( (
f ò F , we get
i # e. Altogether, we proved that M(F ) is a oarser K . In parti ular, is surje tive. show the inje tivity of . Let us take z 1 ; z 2 ò K ; z 1 #Ö z 2 .
onvex ompa ti ation than Now we are going to
Z is a Hausdor lo ally onvex spa e, there exists a linear ontinuous fun tional f0 ò Z that separates the points z1 and z2 . Putting f # f0 K , we obtain f òA (K) su h that f (z1 ) #Ö f (z2 ). Then, for f # f i ò F , we obtain < (z1 ); f> # f (z1 ) #Ö f (z2 ) # < (z2 ); f>. In other words, ( z 1 ) #Ö ( z 2 ), thus is inje tive.
Sin e
*
Eventually, realizing that spa e
is a onetoone ontinuous mapping from a ompa t
K onto a Hausdor spa e M(F ), we an see that also "1 is ontinuous.5
As for the uniqueness of
F
onstru ted above, we have to show that if
F1
and
F2 are two onvexifying, losed linear subspa es of C (U) ontaining onstants su h Ê M(F2 ), then ne essarily F1 # F2 . As there is an ane homeomorphism : M(F1 ) Ù M(F2 ) su h that e1 # e2 , it is easy to see that A (M(F1 )) e1 # A (M(F2 )) e2 be ause obviously A (M(F2 )) # A (M(F1 )), where e l : U ÜÙ Fl* 0
that M(F1 )
l # 1; 2. Simultaneously, we have6 also # A (M(Fl )) e l for l # 1; 2. Altogether, we have obtained
denotes the respe tive evaluation mappings,
Fl
F1
# A (M(F
1 ))
e # A (M(F 1
2 ))
e # A (M(F
2 ))
1
e #F : 2
2
M(F ) onverging weakly* in F to some ò M(F ), then the net { is )} ò K must have a luster point z ò K be ause K is ompa t, but ( z ) # be ause
ontinuous and the weak* topology on M(F ) is a Hausdor one, hen e z is determined uniquely and " ( )} the whole net { ò K onverges to it. f. also, e.g., Engelking [284, Theorem 3.1.13℄. 5
Indeed, taking a net {
"1 (
} ò
1
6
See Berglund at al. [108, Corollary 3.7℄ for details.
*
88
Ë
2 Theory of Convex Compa ti ations
The pre eding theorem authorizes us to de lare (M(F )
; F ; e) with F *
Å a onvex
C0 (U) ontaining onstants as a anoni al form of a onvex ompa ti ation ( K ; Z ; i ) in question whenever (M(F ) ; F ; e ) Ê ( K ; Z ; i ). Let us note that ifying subspa e of
*
this denition does not require
F
to be losed and therefore the anoni al form is not
K ; Z ; i). More pre isely, F C0 (U); f. Theorem 2.8 below. Su h
determined uniquely by a given onvex ompa ti ation ( is determined uniquely only up to the losure in
a denition of the anoni al form ree ts the fa t that in on rete ases we are given by some onvexifying subspa e
F
of
C0 (U) but its losure in C0 (U) usually annot be
ee tively determined. Let us turn our attention to the ordering of onvex ompa ti ations. It is natural to seek still ner and ner onvex ompa ti ations and a natural question in this
ontext is whether there exists maximal onvex ompa ti ations. The following theorem gives an answer in terms of the anoni al form, although, of ourse, by a non onstru tive way via the KuratowskiZorn lemma 1.1.
Theorem 2.7 (Maximal onvexifying subspa es). of
C0 (U)
Any maximal onvexifying subspa e
is losed. Moreover, every onvexifying subspa e
some maximal onvexifying subspa e of
C (U).
F
C (U) is ontained in 0
0
lim"Ù lim ò f " (u ) # lim ò f(u ) proU , that means f " " f C0 U Ù 0, and provided lim ò f " ( u ) does exist for any " ¡ 0. Therefore, if F is onvexifying, in Proof. For any net { u } ò , we have the identity
f
vided a sequen e { " } " ¡0 onverges to
f
0
uniformly on
view of (2.3) we an see that its losure in
(
)
C0 (U) remains also onvexifying. In parti 
ular, no onvexifying subspa e whi h is not losed an be maximal. The rest will be proved by using the KuratowskiZorn lemma. Therefore, we are to prove that, for every olle tion {F } ò A of onvexifying subspa es of
su h
F2 F1 for any 1 ; 2 ò A, there is a onvexifying subspa e 0 F C (U) su h that F F for ea h ò A. We want to show that it su es to take F # U òA F . Su h F is a linear spa e. Indeed, for any f 1 ; f 2 ò F there are 1 ; 2 ò A su h that f1 ò F1 and f2 ò F2 . As F1 F2 or F2 F1 , both f1 and f2 are ontained in F2 or in F 1 , respe tively. Hen e any linear ombination of f 1 and f 2 is ontained in F 2 or in F 1 and, in parti ular, also in F . It remains to show that F is onvexifying. Let us take u 1 ; u 2 ò U and the dire ted that
F1
C0 (U)
F2
or
# (F; n) ò F there is # ( ) ò A su h that F F be ause we an always suppose F # {f i }ki#1 with some k ò N and f i ò F i for some i ò A su h that F 1 F 2 ::: F k , whi h allows us to put simply ( ) # 1 . As F is onvexifying and F is a universal index set ( f. the proof of Proposition 2.5), there is a net { u } ò F U su h that, for every f ò F and n ò N, index set (
# F
as in (2.4). For any
)
(
)
(
(
there is 1
n . As
(
(
)
(
)
£ (f; n) it holds f(u ) " 12 f(u1 ) " 12 f(u2 ) ¢ is dire ted, there is ( F; n ) ò F su h that ( F; n ) £ ( f; n ) for any f ò F ;
# (f; n) ò F
F
)
)
)
su h that for any
(
)
Ë 89
2.2 Canoni al form of onvex ompa ti ations
1
F is nite. Thus we get the situation that f(u ) " 2 f(u1 ) " 12 f(u2 ) ¢ 1n for any £ ( F; n ) and any f ò F . Taking a net { u } ò F and u # u F n for # ( F; n ) and realizing that F F for any ò F , then, for any f ò F , n ò N, £ ({ f } ; n ), we
re all that
(
(
fu
have got ( )
;
)
)
" 21 f(u ) " 21 f(u 1
¢
2 )
Å
1
n . It shows that F is onvexifying.
The following assertion together with Theorem 2.6 show that the topology of the
C0 (U) is de isive for M(F ) in the sense 0 that we an always enlarge F up to its losure in the C topology without any inu
uniform onvergen e on the subspa es
F
of
en e on M(F ). On the other hand, any further enlargement of F will already inuen e
F ). Also note that, sin e F 's are endowed by the (relativized) C0 topology, the embedding F1 F2 is of Type (C) a
ording to the lassi ation from (1.43) whi h auses * * the adjoint operator F2 Ù F1 to be surje tive.
M(
Theorem 2.8 (Dependen e of M(F ) on F ).7 Let F ; F 1
spa es of
2
be two onvexifying linear sub
C0 (U) ontaining onstant fun tions. Then:
F implies M(F ) ² M(F ); i.e. the mapping F ÜÙ M(F ) is monotone. (ii) If l C 0 U F # lC 0 U F , then M(F ) Ê M(F ). (iii) If F F but l C 0 U F #Ö l C 0 U F , then M(F ) ° M(F ). F1
(i)
2
(
1
)
1
1
(
2
2
2
)
(
1
1
)
(
)
2
2
1
2
Proof. It is easy to see that the anoni al ane ontinuous surje tion M(F2 ) xing
Ù M(F
1)
U is just Q : F2 Ù F1 , where Q : F1 Ù F2 is the in lusion F1 F2 . Hen e *
*
*
F2 ) is ner than M(F1 ), as laimed at the point (i). 0 To prove (ii), let us take F a onvexifying subspa e of C ( U ) and put F # lC 0 ( U ) F . * Now let Q : F Ù F be the in lusion F F . As above, the adjoint operator Q : * * F Ù F realizes the anoni al ane ontinuous surje tion M(F ) Ù M(F ) xing U . We want to show that the restri tion of Q* on M(F ) is inversely ontinuous, whi h * will show M(F ) Ê M(F ). As F is dense in F , Q : F * Ù F * is inje tive; indeed, for 1 ; 2 ò F su h that 1 #Ö 2 , the restri tions on the dense subspa e F must M(
Q 1 # 1 F #Ö 2 F # Q 2 . As Q *
*
also dier from ea h other, hen e
*
is a onetoone
ontinuous mapping between two ompa t sets M(F ) and M(F ), the inverse mapping
Ê M(F ). From this ) provided F # F , as laimed
must be also ontinuous.8 Altogether, we have thus proved M(F ) we get immediately M(F1
) Ê M(F ) # M(F ) Ê M(F 1
2
2
1
2
at the point (ii). In the rest of the proof we want to prove that M(F1 )
F2
7
are losed and
F2
The proof of the fa t
of the
C0 (U) losure
of
M(F F
F1
1)
but
F2
° M(F
#Ö
F1 . We want to
° M(F
2 ),
supposing that
prove that there are
F1 ,
1 ; 2 ò
2 ) follows also quite straightforwardly from the uniqueness
for a given onvex ompa ti ation of
U,
f. Theorem 2.6. Nevertheless,
we performed a dire t, hopefully interesting proof whi h is not expli itly supported by Berglund at al. [108, Corollary 3.7℄ used for the uniqueness in Theorem 2.6. In fa t, [108, Corollary 3.7℄ relies on similar arguments as HahnBana h, Riesz and Jordan theorems used in the proof presented here.
8
Let us note that the embedding is of the type (AB) a
ording to the lassi ation on p. 36.
90
Ë
2 Theory of Convex Compa ti ations
M(F2 ) su h that 1 #Ö 2 but < 1 ; f > # < 2 ; f > for any f ò F1 , whi h shows that Q 1 # Q 2 for Q : F1 Ù F2 being the in lusion F1 F2 , hen e M(F1 ) ÊÖ M(F2 ). *
*
As M(F1 )
² M(F
2)
° M(F
has been already proved, this will imply M(F1 )
Let us denote by F
1
the annihilator of
F1
2 ).
*
F1 in F2 , this means9
# ò F ; :f ò F :
; f
*
1
2
# 0:
We show that M(F2 ) is not at in ea h dire tion from F1 in the sense that, for any
ò
F1 dierent from zero, there are 1 ; 2 ò M(F2 ) su h that 1 " 2 # with some #Ö 0. Indeed, ò F means, in parti ular, that < ; 1> # 0. Let U be a ompa ti ation of U 1
f ò F admits a ontinuous extension f on U . Then f ÜÙ forms a linear ontinuous fun tional on F # {f ; f ò F } C( U). By the
su iently ne10 su h that every
2
HahnBana h theorem 1.11, this fun tional admits the ontinuous linear extension on
C( U), and then by the Riesz representation theorem 1.32(iii) f d for any f ò F2 . Then we an there is a measure òr a( U ) su h that < ; f > # P U
the whole Bana h spa e
make the Jordan de omposition of
%
onto its positive variation
and its negative
* % " " variation , whi h again belong to r a( U ). Then we dene ; ò F2
for any
f ò F2 .
%; f
#
X
f d %
and
U
Therefore we an see that
"; f
#
X
by
f d "
U
admits a (generally not uniquely deter
% " " with both %
# £ 0 and " £ 0. We want to prove that " # ¡ 0. Supposing the ontrary, both % and " would be % " identi ally zero sin e the orresponding measures and would be nonnegative % with < ; 1> # P d % # 0 and
zero. Hen e also
# 0, but we supposed #Ö 0, a ontradi tion. Therefore we may put 1 #
Obviously,
2 ò M(F2 ).
1 £ 0
For any
and
and
2 #
" : " < ; 1>
1 ; 1> # 1, hen e 1 ò M(F2 ),
see (2.1), and similarly also
f ò F2 \ F1 there is some f ò F1 su h that #Ö 0 be ause otherwise f
would have to belong to the annihilator of
F1 in F2 , whi h is the losure of F1 , hen e
# f and making the above des ribed de omposition, we obtain the situation < ; f > #Ö < ; f > for f ò F \ F , whi h shows that a tually #Ö , but, on the other hand, < ; f > " < ; f > # < ; f > # 0 for any f ò F . Å F1 itself. Choosing 1
2
1
2
2
1
1
2
1
F1 #Ö {0} be ause otherwise F2 F1 , ontrary to what is supposed. 10 For example, if U is ompletely regular, one an always take for U the e hStone ompa ti ation U . If not, one an rene the topology on U be ause, e.g., a dis rete topology is ertainly om9
Let us noti e that
pletely regular.
Ë 91
2.2 Canoni al form of onvex ompa ti ations
Corollary 2.9 (Maximal onvex ompa ti ations).
For every onvex ompa ti ation
K of U , there exists a maximal onvex ompa ti ation K1 of U ner than K . Proof. By Theorem 2.6, we an take a losed onvexifying linear subspa e
F
of
C0 (U)
K Ê M(F ), and by Theorem 2.7 there is a maximal C (U) su h that F F . Let us put K # M(F ). K ² K as a onsequen e of Theorem 2.8. To show that
ontaining onstants su h that
onvexifying subspa e
F1
0
F , it holds ( K ; Z ; i ) # (M(F ) ; F ; e F
As
1
1
1
1
*
1
1
1
1
1
1
1)
is maximal, let us take another onvex ompa ti a
K2 ; Z2 ; i2 ) of U su h that K2 ³ K1 . We want to demonstrate that K2 ² K1 . Let us put F2 # A ( K 2 ) i 2 and F1 # A ( K 1 ) i 1 . Then we have K 2 Ê M(F2 ) and K 1 Ê M(F1 );
tion (
f. the proof of Theorem 2.6. On the other hand, we have also
# A (M(F )) e surje tion :K ÙK
F1
1.
1
2
1
Sin e we supposed
i # i
su h that
2
1.
K2 ³ K1 ,
F1
#
F1
be ause11
there is an ane ontinuous
Then obviously
A (K ) A (M(F
2 )).
1
Altogether, we have obtained
However, Hen e
F1
#
F1
was a maximal onvexifying subspa e
F1
# A (K ) i # A (K ) 1
1
1
i A (K ) i #
K2 Ê M(F2 ) # M(F1 ) # K1 has been proved.
2
2
F1
2
F2 :
C (U), so that F # 0
2
F1 .
Å
Now we want to relate the standard onstru tion of a ompa ti ation by multipli ative means with the onvex ompa ti ations. Every ompa ti ation, reated from a onvex ompa ti ation by forgetting the lo ally onvex spa e, must be obviously equivalent in the usual sense to some ompa ti ation obtained by the multipli ative means. Considering a onvex ompa ti ation in its anoni al form (M(
F ); F * ; e), as we always an due
to Theorem 2.6, we want now to onstru t the
equivalent standard ompa ti ation expli itly. Besides, we shall see that the homeomorphism that makes them equivalent to ea h other is even ane. Let us denote by
Ring(F ) the smallest losed subring of C (U) ontaining F C (U), i.e. 0
n
0
m
Ring(F ) # lC0 U H I f ij ; n ; m ò N; f ij ò F : (
)
(2.6)
i #1 j #1
Theorem 2.10 (Conne tion with multipli ative means). Let F be a linear subspa e of C (U) ontaining onstant fun tions and Ring(F ) the orresponding smallest losed ring ontaining it, and let Q : F Ù Ring(F ) denote the in lusion F Ring(F ). Then: (i) The adjoint operator Q realizes a homeomorphi al embedding of Mmult (Ring(F )) 0
*
into M(F ). (ii)
Q
*
(Mmult (
Ring(F ))) #
F ), i.e. Q*
M(
is a homeomorphism, if and only if
vexifying.
11
See Berglund at al. [108, Corollary 3.7℄ for details.
F
is on
Ë
92
Q
Proof. Mmult (
2 Theory of Convex Compa ti ations
*
is
Ring(F )))
obviously
ontinuous
and
maps
Ring(F ))
(and
M(
thus
also
Q
into M(F ). Now we will prove the inverse ontinuity of
Ring(F )), this means:
*
if
restri ted on Mmult (
: ò Mmult (Ring(F )) :" ¡ 0 :f òRing(F ) ;Æ ¡ 0 ;k ò N ;{f l }kl# F 1
: ò Mmult (Ring(F )): max Q ; f l " Q ; f l ¢ Æ á *
*
¢l¢k
1
!! ! !! ; f " ; f !!!
¢ ":
(2.7)
Ring(F ), there exist m ; n ò N and f ij ò F , i # 1; :::; n, j # 1; :::; m, n m f " ni#1 A m j #1 f ij C 0 U ¢ " /3. Then also < ; f > " < ; i #1 A j #1 f ij > ¢ " /3 and n m < ; f > " < ; i #1 A j #1 f ij > ¢ " /3 be ause # # 1. Now we an estimate:
By the denition of su h that
(
)
!! !! ; f
!!
" ; f !!!! ¢ !!!! ; f " ¼ ;
!
%
n m n m !! ! !!¼ ; H I f ½ " ¼ ; H I f ½!!! % ; ij ij !! !! ! ! i #1 j #1 i #1 j #1
1 " % 3
¢
n
n m !! ! H I f ij ½!! !! i #1 j #1
n ! m m !! ! H !!! I ; f ij " I ; f ij !!! !! !! i #1 j #1 j #1
m
¢ L H H !!!! ; f ij " ; f ij !!!! %
i #1 j #1
%
n m !! !!¼ ; H I f ½ ij !! ! i #1 j #1
!!
" ; f !!!!
!
1 " 3
2 "; 3
L is the Lips hitz onstant of the mapping (a1 ; :::; a m ) ÜÙ A m j #1 a j restri ted to the arguments a j with a j ¢ max i j f ij C 0 U . Using the fa ts that < Q ; f ij > # < ; f ij > and < Q ; f ij > # < ; f ij > be ause f ij ò F , we obtain (2.7) when taking Æ # " /(3 nmL ), k # nm, and {f l } # {f ij }. Thus the point (i) has been proved. Now suppose that Q is the homeomorphism of Mmult (Ring(F )) onto M(F ). Let R : U Ù M(Ring(F )) from the evaluation us distinguish the evaluation mapping e R R mapping e : U Ù M(F ). Clearly, e # Q e . Sin e e ( U ) is dense in Mmult (Ring(F )), e(U) is dense in Q (Mmult (Ring(F ))) # M(F ). By Proposition 2.5 (ii) á (i), F must be
where
*
;
(
)
*
*
*
*
then onvexifying. It remains to prove the onverse impli ation, supposing that
F
is onvexifying.
á (ii), we an see that e(U) is dense in M(F ). By the obvious estimate e ( U ) Q (Mmult (Ring (F ))) M(F ) and by the weak* ompa tness of Mmult (Ring(F )) and the weak* ontinuity of Q , we get eventually the surje tivity of Q : Mmult (Ring(F )) Ù M(F ): Å Using Proposition 2.5 (i) *
*
*
Let us remark that the adjoint operator
Q : (Ring(F )) Ù F
ing theorem makes nothing else than the restri tion on fun tionals
Ring(F ) Ù R.
*
*
F
*
from the pre ed
of the linear ontinuous
Theorem 2.10 showed M(F ) equivalent (as a usual ompa ti ation) with
Ring(F )).
Mmult (
However, it should be pointed out that Mmult (
Ring(F ))
denitely
2.3 Convex
Ë 93
 ompa ti ations
annot serve well as far as onvexity on erns be ause it is extremely bent in
Ring(F ))
(
*
, whi h is made pre ise by the following assertion.
Proposition 2.11.
Let F be a subspa e of C ( U ) ontaining onstant fun tions, 0
the losed subring generated by
Ring (F ) % 12 ò
1 F , and let 1 ; 2 ò Mmult (Ring(F )). Then 2 1
Ring(F )) \ Mmult (Ring(F )) whenever #Ö
M(
1
2
2.
1 # 21 1 % 2 2 ò Mmult (Ring(F )). In other # for all f1 ; f2 ò Ring(F ). Taking f # f1 # f2 ò Ring(F ),
Proof. Suppose the ontrary, that means words,
we get
1 1 1 1 2 2 2 2 2 2 ; f % 2 ; f # 1 ; f % 2 ; f # ; f # ; f 2 1 2 2 2 2 # 12 1 ; f % 12 2 ; f # 41 1 ; f 2 % 14 2 ; f 2 % 12 1 ; f 2 ; f :
1 ; f>2 % 2 # 2, whi h means pre isely # < 2 ; f >. As f òRing(F ) is arbitrary, we have obtained 1 # 2 , a ontradi tion. Å
This is true only if
# P g ( x ) v 0 ( u ( x ))(1% u ( x ) ) d x . In other words,
p m this means < i ( u ) ; h > # P h ( x ; u ( x ))(1% u ( x ) ) d x for h ò C ( , R R ). We onsider
m weak* topology on r a( , R R ), whi h makes i ontinuous. Thanks to the estimate *
embedding
*
i u ) r a ,
(
(
R
¢ % u pLp Rm ;
Rm )
(
;
(3.48)
)
òr a( ,
iu
the net { ( )} ò is bounded, and it must have a luster point
R
Rm . There)
fore there is a ner net (denoted for simpli ity by the same indi es, f. Example 1.4)
1%u p } ò , being bounded in L ( ), has a
onverging weakly* to . Besides, the net {
luster point in
1
r a( ), say , and we may and will assume that our ner net has been
hosen so that (3.46a) holds, as well. Let us dene
T : R Ù r a( ) Ê C( )
*
T v0 ; g
for any
by
# ; g v
0
g ò C( ). In view of (3.48), we have the estimate
!! ! !! Tv 0 ; g !!!
¢ v
0
#
C 0 (Rm )
!! ; g !!
v
0
!
!!!
!!
!!
# !!!! lim X g(x)v (u (x))(1% u (x)p ) dx!!!! 0
! ò
lim X g(x)1%u (x)p dx # v ò
!
0
C 0 (Rm ) X
g(x)(dx):
(3.49)
T v0 ℄(A) ¢ v0 C0 Rm (A) for any Borel subset A , whi h , and thus by the Radon1 Nikodým theorem it admits a representation by a density Tv 0 ò L ( ; ), that means 1 [ T v 0 ℄( A ) # P [ Tv 0 ℄( x ) (d x ). The mapping T : R Ù L ( ; ) is linear, and in fa t it is A
In parti ular, it implies [ means that
Tv0
(
)
is absolutely ontinuous with respe t to
Ë
148
3 Young Measures and Their Generalizations
T:RÙL (3.49) and of the duality between L ( ; ) and L ( ; ). bounded (with the norm being equal 1) as an operator 1
(
; ) be ause of
Obviously, (3.46b) follows easily from the denitions, namely
X g ( x ) v 0 ( u ( x ))(1% u ( x )
p
)
d x # i ( u ) ; g v # Tv ; g #
Ù ; g v
X
0
R R
0
0
g(x)[Tv0 ℄(x)(dx) :
R
0 m ) is separable, hen e m is metrizable. As Suppose now that R C ( R m is separable and dense in m , we have m separable. As is regular, R R deC( ) C( R m ) is dense in L1 ( ; ; C( R m )).40 Therefore, the fun tional 1 m ned above an be extended ontinuously onto L ( ; ; C ( )). By DunfordPettis R 1 m ))* Ê L ( ; ; r a( m )), whi h assigns an element theorem, L ( ; ; C ( w R R m )). ò L ( ; ; r a( w* R m and repla ing the Lebesgue meaIt is easy to verify (3.17) for with S # R p m £ 0. Also, as sure. Indeed, as i ( u ) £ 0 for any u ò L ( ; ), in the limit we get p ; g 1>. Simultanealways 1 ò R , we have P g ( x )(1% u ( x ) ) d x # < i ( u ) ; g 1> Ù <
p ously, P g ( x )(1% u ( x ) ) d x Ù P g ( x ) (d x ), whi h shows that P P
A Rm x (ds)(dx)
R
R
R
R
R
R
R
R
R R
# (A) for any Borel subset A
, R Rm Ù .
R
or, in other words,
is the proje tion of via
S
Then by Lemma 3.5, whi h holds true not only for able ompa t
S#
R
R
m , we obtain that ò Y( ; ;
R
R
Rm but also for the metriz
m ). Moreover, for any g ò C ( ),
it holds
X X
R
Rm
g(x)v0 (s) x (ds)(dx) #
()
; g v0
# ; g v
0
#
X
g(x)[Tv0 ℄(x)(dx) ;
Lw ( ; ; r a( R Rm )) Ù L1 ( ; ; C( also Lemma 3.4. As (3.50) holds for any g , (3.47) has been proved.
where
denotes the isomorphism
Let us agree to denote the set of all Radon measures on
,
R
R
(3.50)
Rm
*
)) , f.
Å
Rm onstru ted in
the previous proof by
p m òr a( , R R m ); DMR ( ; R ) # ;{u } ò bounded in L p ( ; Rm ) :
If
R
40
C
0
Rm
(
w*
) is separable, we will also work with the set
See Warga [791, Thm. I.5.25℄.
lim i(u ) # Ǳ: ò
(3.51)
3.2 Various generalizations
DMR ( ; Rm ) # ( ; ) òr a% ( ) , Y( ; ; p
;{u } ò bounded in L
p
(
Rm ;
; Rm : )
R
(3.46a)(3.47) holdǱ
)
Let us agree to address the elements of both
Ë 149
p m DMR ( ; R )
and
:
(3.52)
DMR ( ; Rm ) p
as
DiPernaMajda measures. It should be emphasized that is not a Young measure onstru ted in Theorem 3.19. A tually, the relation between the DiPernaMajda and the Young measures is stated in the following assertion. Let us note that the ontributions of a given DiPernaMajda measure supported at innity (this means on the remainder
Rm Rm ) are
R p "1 used in (3.54) vanishes at innity. forgotten be ause the weight (1% s )
Theorem 3.26 (DiPerna and Majda).41 Let {u k }kòN L p ( ; Rm ) and R a separable omplete subring of C
0
be
a
(
)
bounded
Rm , p ò
sequen e
1; %).
[
\
in
Then there
is a subsequen e, denoted again by { u k } k òN , onverging to a DiPernaMajda measure
(
; ) ò DMR ( ; Rm ) in the sense p
lim X g(x)v(u k (x)) dx k Ù
for any
g ò C( )
and any
#
X X
R
Rm
v(s) # v0 (s)(1%sp )
g(x)v0 (s) x (ds)(dx) ;
with
v0 ò R.
sequen e onverges in the sense of (3.34) to a Young measure
x (d s )
# d (x)
(3.53)
Simultaneously, this sub
ò Yp ( ; Rm ) given by
x (d s ) 1%sp
(3.54)
is absolutely ontinuous (with respe t to the Lebesgue measure on ) with d being its density.42
provided
Proof. Obviously, (3.53) is a mere ombination of (3.46b) and (3.47) together with the fa t that the separability of
R
R
R
implies metrizability of the weak* topology of
C( ,
m )* restri ted on bounded subsets, whi h eventually allows us to work in terms
of subsequen es. Let us show that
from (3.54) is a tually a parametrized probability measure (i.e
a Young measure). Let us put
X g(x) dx
#
v0 (s) # (1%sp )"1 into (3.53). Realizing that v # 1, we get
X g ( x ) v ( u k ( x )) d x
Ù
X g ( x ) ¤X
x (d s ) R
Rm
1%sp
¥ d (x) dx ;
whi h implies
41
We refer to DiPerna and Majda [266, Corollary 4.3℄. The assertion here is, however, slightly gener
42
In fa t, formula (3.54) holds even if
absolutely ontinuous part of
R
L ( ; m ) has been admitted for (3.54) in [266℄. is not absolutely ontinuous then d is the density of the
alized be ause only sequen es bounded in
in its Lebesgue de omposition.
150
Ë
3 Young Measures and Their Generalizations
x (d s )
X R
for a.a.
1%sp
Rm
#1
d (x)
x ò . Realizing that s ÜÙ (1%sp )"1 vanishes on the remainder
obtain
R
Rm Rm , we \
x (d s ) x (d s ) # d (x)X # 1; m 1%sp 1%sp RR x is a probability measure for a.a. x ò be ause the fa t that
x (d s )
X
Rm
whi h shows that
# d (x)X
Rm
x is
nonnegative is lear from its denition (3.54).
u
Simultaneously, we know from Theorem 3.19 that a (sub)sequen e { k } determines a Young measure, let us denote it by
1
oin ides with
. It remains to show that
g ò C( ) and v ò C p (Rm ), we know from (3.34) and (3.53) that P g(x)v(u k (x)) dx 1
onverges to P P m g ( x ) v ( s ) x (d s ) d x and to P P Rm g(x)v0 (s) x (ds)(dx), respe R
1
. For
R
tively. Therefore, we have
X X
Rm
x (d s ) d x
g(x)v(s)
#
1
#X
#X
X R
X
Rm
Rm
g(x)
g(x)
X X
R
v(s)
1%sp
v(s)
1%sp
Rm
g(x)v0 (s) x (ds)(dx)
x (d s ) d ( x ) d x
x (d s ) d ( x ) d x
#X
X
Rm
Of ourse, we used the denition (3.54) and the fa t that remainder
R
g(x)v(s)
x (d s ) d x :
v(s)/(1% sp ) vanishes on the
Rm Rm . Sin e g and v have been taken arbitrarily, the identity \
1
#
is proved.
u
For illustration, let us give examples of some sequen es { k }
Å
L p ( ; Rm ) and the
orresponding DiPernaMajda measures ( ; ) generated by them. As a rst, trivial p m ) and onsider # i ( u ) òr a( , R m ). example, let us have a fun tion u ò L ( ; Then the orresponding DiPernaMajda measure ( ; ) takes the form
R
R
(dx) # 1%u(x)p dx ;
x
# Æu x ; (
(3.55)
)
and we an see that (ex eptionally!) oin ides with the orresponding Young measure when one repla es
by the Lebesgue measure.
A bit less trivial example is the situation outlined on Figure 3.3, namely for two
u1 ; u2 ò L p ( ; Rm ) let us onsider an os illating sequen e { u k } L p ( ; Rm ) m # ( i ( u 1 ) % i ( u 2 ))/2 ò r a( , whi h onverges weakly* to R R ), and the orresponding Young measure is x # ( Æ u 1 x % Æ u 2 x )/2. In view of (3.46a) and (3.54), we fun tions
( )
(
)
an easily establish the on rete form of the orresponding DiPernaMajda measure (
; ), namely
1 p p (2 % u ( x ) % u ( x ) ) d x ; 2 1%u (x)p 1%u (x)p # Æ u1 x % Æ : p p 2 % u (x) % u (x) 2 % u (x)p % u (x)p u2 x
(dx) # x
1
2
1
2
(
1
2
)
(
1
2
)
/ 7
(3.56) ? 7 G
3.2 Various generalizations
All the rest examples will use
R %
Ë 151
# (0; 1), m # 1, and R from (3.45) so that
R Ê 1
R
; "}. They deal with the situations when some nonvanishing part of the u p is a tually arried to innity. The righthand part of these gures is to ) , R R1 # [0 ; 1℄ , (R {% ; "}). Let us note that in ea h ase indi ate supp( the orresponding Young measure ò Y((0; 1); R) is homogeneous, namely x # Æ0 for a.a. x ò (0 ; 1). p The example shown in Figure 3.7 presents a sequen e { u k } k òN su h that u k onp
entrates, onverging just to a Dira distribution (multiplied by the fa tor a b ) at a point x 0 ò . The resulting DiPernaMajda measure is given by {
energy k
# 1 % a p bÆ x0 ; where
x
#
Æ0 Æ%
if if
is determined obviously from (3.46a) while
x #Ö x 0 ; x # x0 ;
(3.57)
is determined from (3.53) with
help of the identity
lim
1
k Ù
X 0
g(x)v(u k (x)) dx # lim
x 0 " b/(2 k ) X g(x)v0 (0) dx
k Ù 0 x 0 % b/(2 k ) %X g(x)v0 (ak1/p )(1 x 0 " b/(2 k )
repla ements 1
% a p k) dx % 1
# X g(x)v (0) dx % a p bg(x )v (%) # X X 0
0
0
0
S
1
0
%
uk
1
g(x)v0 (0) dx X x 0 % b/(2 k ) g(x)v0 (s) x (ds)(dx):
R{%;"}
%
ak1/p
k
Ù
b/k
0 Fig. 3.7:
0
x0
x0
The DiPernaMajda measure reated by a on entrating sequen e.
The next example shows a homogeneous DiPernaMajda measure (i.e. both
do not depend on
and
x) reated by an os illating sequen e s hemati ally outlined on ; ) is the following:
Figure 3.8. The resulting DiPernaMajda measure (
(dx) # 1 % a p b "1 dx
and
x
#
ap b Æ0 % Æ p
%a b
% ap b %
;
(3.58)
whi h an be determined respe tively from (3.46a) and (3.53) similarly as in the pre eding example.
Ë
152
PSfrag repla ements
S
1
3 Young Measures and Their Generalizations
% u k
%
ak1/p
k
Ù
b/k2
0
c k
Fig. 3.8:
0
2c 3 c k k
The DiPernaMajda measure reated by an os illating sequen e.
The last example shows a DiPernaMajda measure reated by an os illating sequen e whose os illations on entrate near a point DiPernaMajda measure (
; ) is the following:
# 1 % 3a p bÆ x0
and
x
#
Æ0 2 Æ% 3
x0 ò ; f. Figure 3.9. The resulting
if
% Æ" 1
3
if
x #Ö x 0 ; x # x0 :
(3.59)
PSfrag repla ements
S
1
%
uk
%
ak1/p
uk
k
0
x0
Ù
0
x0
b/k
"ak
1/
p
" Fig. 3.9:
" The DiPernaMajda measure reated by an os illating/ on entrating sequen e.
Let us noti e that the DiPernaMajda measures ( possess the omponent
; ) from (3.55), (3.56), and (3.58)
absolutely ontinuous, while those from (3.57) and (3.59) do abso
not. Obviously, (3.58) demonstrates that even DiPernaMajda measures with
lutely ontinuous an still arry some part of energy to innity. Therefore, a stronger kind of regularity is still worth to be onsidered: let us all a DiPernaMajda measure (
; )
to be
onverging to (
;
pnon on entrating
weakly ompa t in
L1 ( ).
u
if there is a bounded net { } ò
) in the sense of (3.53) su h that the set
{
u
p ;
L p ( ; Rm )
ò } is relatively pnon on
Let us only remark that the property of being
entrating is quite natural be ause solutions of oer ive relaxed optimization prob
Ë 153
3.2 Various generalizations
pnon on entrating,
lems are typi ally
as we will see later in Propositions 4.46(iv),
4.76(iv), 4.116(iv), 5.21 and 7.15(i), and Remark 6.23.
Lemma 3.27 (Non on entrating DiPernaMajda measures). Let the omplete R C (Rm ) be separable and ( ; ) be a DiPernaMajda measure. Then:
subring
0
(i)
(
; ) is pnon on entrating if and only if its energy is not supported at innity
(i.e. on the remainder
R
Rm Rm ) in the sense that \
X X
R
Rm \Rm
x (d s ) (d x )
# 0:
(3.60)
; ) is pnon on entrating, then is absolutely ontinuous and any bounded sep m quen e { u k } k òN L ( ; R ) onverging to ( ; ) does not on entrate energy (i.e. p 1 the set { u k ; k ò N} is relatively weakly ompa t in L ( )) and the orresponding
(ii) If (
Young measure onstru ted in Theorem 3.26 is fully ee tive in the sense that (3.34)
v ò C p (Rm ).
holds even for any
L p ( ; Rm ) be a bounded sequen e onverging to ( ; ) in the
Proof. Let { u k } k òN sense of (3.53).
u
First, let us suppose that { k
p;
N
kò
} is relatively weakly ompa t in
L1 ( ).
1%u k k ò N}, and by Theorem 1.28(ii) it is also uniformly :" ¡ 0 ;r " ò R% : p ;
Therefore, so is also set {
integrable, whi h means:
sup X k òN
For
{
x ò ; u k ( x )£ r " }
r ¡ 1, let us dene v0r # 1 " v r
with
v0 ò R , so that we an estimate r
X X
m m R R \R
x (d s ) (d x )
¢
1%u k (x)p dx ¢ " :
vr
dened by (3.38). Let us note that always
X X
m RR
v0r " (s) x (ds)(dx)
r" p X v 0 u k ( x ))(1% u k ( x ) d x k Ù
# lim
¢ sup X k òN
As
1%u k (x)p dx ¢ " :
{
x ò ; u k ( x )£ r " }
" ¡ 0 was arbitrary, (3.60) has been proved. Let us prove the onverse impli ation. Supposing the DiPernaMajda (
es (3.60) and putting
lim X X r Ù%
R
B r # {s ò Rm ; s ¢ r}, we get Rm \ B r
x (d s ) (d x )
#
X X
R
Rm \Rm
x (d s ) (d x )
# 0;
m òr a( , additivity of the measure RR ) Theorem 3.25. Let us now take " ¡ 0. Then for r
whi h follows simply by means of the assigned to (
; );
f. the proof of
su iently large we have got
X X
R
Rm
; ) satis
v0r (s) x (ds)(dx) ¢
X X
R
Rm \ B r
x (d s ) (d x )
¢
"
2
:
Ë
154
3 Young Measures and Their Generalizations
Moreover, there is some
!! !!X X !!!
m RR
k r ò N su h that, for every k £ k r , !! r p X v 0 ( u k ( x ))(1% u k ( x ) ) d x !!! !!
v0r (s) x (ds)(dx) "
Altogether we obtained P
¢
"
2
:
v0r (u k (x))(1%u k (x)p ) dx ¢ " for any k £ k r , and therefore
also
1%u k (x)p dx ¢
X
x ò ; u k ( x )£ r %1} p The nite set {1% u k ; k {
L1 ( ),
P
{
# 1; :::; k r }
r p X v 0 ( u k ( x ))(1% u k ( x ) ) d x
is obviously relatively weakly ompa t in
hen e uniformly integrable, whi h means that for some
x ò ; u k ( x )£ r 0 }
¢ ":
r0
su iently large
1%u k (x)p ) dx ¢ " for any 1 ¢ k ¢ k r . Altogether, we got for any k ò N
(
X {
x ò ; u k ( x )£max( r 0 ; r %1)}
1%u k (x)p dx ¢ " :
" ¡ 0 was arbitrary, we have proved that the whole sequen e {1%u k p }kòN is uni1 formly integrable, hen e also relatively weakly ompa t in L ( ), so that, by the denition, the DiPernaMajda measure ( ; ) is p non on entrating. This nishes the point
As
(i).
; ) is pnon on entrating, it must full (3.60) and then we saw in the pre eding part of the proof that any { u k } k òN onverging to ( ; ) p in the sense of (3.53) has the property that { u k ; k ò N} relatively weakly ompa t in L1 ( ). In parti ular, also {1%u k p ; k ò N} enjoys this property. In view of (3.46a), we p 1
an see that the limit of {1% u k } k òN is in L ( ), hen e is absolutely ontinuous. Fi1 nally, let us observe that { v ( u k ); k ò N} is relatively weakly ompa t in L ( ) provided p m v ò C (R ). Then (3.34) follows by Lemma 3.20. Å Let us go on to the point (ii). Sin e (
If the ring
DMR ( ; Rm ) p
R
is separable, even a omplete hara terization of elements of
an be established. We present it here rather for an illustration how
parti ular generalizations of lassi al Young measures an be, in fa t, fairly ompli ated.
Proposition 3.28 (Chara terization of DiPernaMajda measures).43 Let the ring R be m ) ò r a( ) , L w ( ; ; r a( R R )). Then the following two state
separable and ( ;
*
ments are equivalent with ea h other: (i)
The pair ( ; ) is a DiPernaMajda measure, i.e. ( ; )
(ii)
dened by (d x ) # P m , and R
p m ò DMR ( ; R ).
x (d s )
(d x ) satisfy
òr a% ( ) ;
(3.61a)
43
Let us note that the absolute ontinuity of
on the remainder
R
R
m
\
R
m
provided
, laimed in
on entrates at the point
the oarsest (i.e. Alexandro 's onepoint) ompa ti ation of DiPerna and Majda [266, Formula (4.18)℄.
(3.61 ), for es
Rm
x
x
to be fully supported
in question. For the ase of
, this observation has been made by
3.2 Various generalizations
R
òr a% ( )
is absolutely ontinuous and
: where
d
Rm
ò Y( ; ;
a.a.
xò :
:
and
)
d (x) #
a.a.
xò :
Rm
Rm
x (d s )
¥
1%sp
¡ 0;
(3.61b) (3.61 )
"1
x (d s )
¤X
X
Ë 155
X
Rm
x (d s ) ;
(3.61d)
with respe t to the Lebesgue measure. is the density of
á(ii). The rst
Proof. (Partly by Kruºík [457, 458℄.) Let us start with the impli ation (i)
part of (3.61b) with (3.61a) has been already shown in Theorem 3.26. To show that
P m x (d s ) R
¡ 0 for a.a. x ò , let us realize that the Lebesgue measure is absolutely
ontinuous with respe t to , having the density d ò L ( ; ) given by44 1
d (x) # P dx Ai
# PA i PRm
x (d s ) (d x )
:
(3.62)
A # {x ò ; PRm
x (d s )
# 0: This proved (3.61b).
ò Yp ( ;
Let us take a Young measure
1%sp
Rm
This density inevitably vanishes on the set
p "1 (1% s )
x (d s )
X
Rm
# 0} so that A #
) generated by a sequen e whi h attains
; ) in question and dene the absolutely 1 measure by means of the density d ò L ( ) given by
the DiPernaMajda measure (
d (x) # 1 % X
Rm
Using (3.34) and (3.53) with passage for
rÙ
s
p
ontinuous
x (d s ) :
v(s) # (1%sp )v r (s) with v r from (3.38) and making a limit
, we obtain by the Lebesgue dominated onvergen e theorem 1.30
the identity
(A) # A % X
X
Rm
A
for any Borel subset
s
x (d s )d x
p
#X
A
X
Rm
x (d s ) (d x )
# (A)
. In parti ular, A , whi h shows that #
is absolutely
ontinuous, as laimed in (3.61 ). Denoting the Lebesgue measure by Nikodým derivative45
d #
By the very denition of , we have
P m R
p "1 (1% s )
0 ,
we an use the formula for the Radon
d d d # : d d d 0
(3.63)
0
d /d # PRm x (ds). By (3.62), we have d /d # 0
x (d s ). Plugging it into (3.63) just gives (3.61d).
à(ii). Let us put
Let us go on to the onverse impli ation, i.e. (i)
x (d s )
# d (x) X
44
This formula follows from (3.53) with
45
See Halmos [374℄ for details.
Rm
v # 1.
x (d s )
"1 [ m ℄(ds) x R
1%sp
:
(3.64)
Ë
156
3 Young Measures and Their Generalizations
"1
# d (x) PRm x (ds)
PRm (1%sp )" x (ds) # 1 as a dire t onsequen e of (3.61d). Also, x ÜÙ x is weakly measurable46 so that ò Y( ; Rm ). Moreover, PRm sp x (ds) # PRm (1%sp ) x (ds) " 1 # " d (x) PRm x (ds)
PRm x (ds) # d (x) " 1, whi h belongs to L ( ) as a fun tion p m of x . Therefore even ò Y ( ; R ). x (d s )
x is positive and PRm
Obviously,
1
*
1
Besides,
; ) if tested by fun tions p. Indeed, for g ò C( ) and v ò C p (Rm ) one has
gives the same result as (
stri tly less that
X X
Rm
1
with a growth
x (d s ) d x
g(x)v(s)
#X
g(x)
X
Rm
#X
X
Rm
v(s)
g(x)
d (x) P m R
v(s) x (d s ) d x x (d s ) 1% s p
x (d s ) (d x )
1%sp
#X
X R
Rm
g(x)v 0 (s) x (ds)(dx)
v 0 ò C( R Rm ) being a ontinuous extension of v0 ò C0 (Rm ) dened by v0 (s) # v(s)/(1%sp ); note that v 0 vanishes on the remainder R Rm \ Rm . Let us now take the sequen e { u k } k òN onstru ted in Proposition 3.22, i.e. it generp 1 and simultaneously the set { u k ; k ò N} is relatively weakly ompa t in L ( ); ates
with
f. Remark 3.23. Our aim is now to modify this sequen e suitably to attain the original
; ). both
DiPernaMajda measure ( As
R
is separable,
therefore, for every
Pl #
{
j1 l V
j J(l) l } j #1 of
j2 l # i diam( ) l
m
and
R
Rm Rm \
are metrizable ompa t sets and
N, there exist nite partitions Pl # il Ii#l of
Rm su h that il1 V il2 # ; 1 ¢ i i ¢ I l
l ò
R
R
{
\
1
}
( )
1
and
( ) and
2
j l are measurable with
; 1 ¢ j1 j2 ¢ J(l) 1/l and diam( lj ) 1/l for all i and j. Besides, we may suppose that, for any l ò N, the partitions P l %1 and P l %1 are respe tively renements of the partitions P l i and P l and that int( ) #Ö for all i . We shall denote by v 0 the ontinuous extension l m m of v 0 ò R on R R , i.e. v 0 ò C ( R R ). We an dene i and moreover all and l
a lij # X
il
Let us hoose
X
,
46 47
R
Rm
1 ¢ i ¢ I(l); 1 ¢ j ¢ J(l):
x ij ò int( il ), 1 ¢ i ¢ I(l), 1 ¢ j ¢ J(l), x ij1 #Ö x ij2 , j1 #Ö j2 , s j ò
òr a( ,
l a measure
X x (d s ) (d x ) ; j l
R
Rm
I(l) J(l)
X
v0 (s) (ds)g(x) (dx) % H H v 0 (s j )g(x ij )a lij
Rm
i #1 j #1
In view of (3.64), this follows from the measurability of
R
C( ); v0 ò R } is dense in C( ,
l j and dene
) by the following formula47
g(x)v 0 (s) l (dxds) # X
We used also the fa ts that
(3.65)
m
is a Borel subset in
m RR
).
d
R
R
and the weak
m
*
(3.66)
measurability of
and that the linear hull {
g
.
v ; g ò 0
3.2 Various generalizations
for any
,(
l g ò C( ) and any v0 ò R . In other words,
aggregates the part supported on
R
R
m
\
Rm
Ë 157
) so that it has the form
I(l) J(l)
l l #
,Rm % H H a ij Æ i #1 j #1
(
x ij ; s j )
;
òDMR ( ; Rm ) orresponds to ( ; ) ò DMR ( ; Rm ) in question. m is dense in m Now let us take l ò N xed. As R R R , for any j there is a sek k m m su h that lim quen e { s } k òN R k Ù s j # s j in R R , whi h means pre isely j p
where
p
limkÙ v (s kj ) # v (s j ) for any v ò R . Inevitably, limkÙ s kj # %. We an k dene neighbourhoods N of points x ij for k ò N, 1 ¢ i ¢ I ( l ) and 1 ¢ j ¢ J ( l ) by ij k l N ij # x ò ; x " x ij (a ij /s kj p B(1)) n where B(1) is the Lebesgue measure of the k k n unit ball in R . Let us note that, sin e s Ù % for k Ù , N are pairwise disjoint j ij k i and N , 1 ¢ i ¢ I ( l ) and 1 ¢ j ¢ J ( l ) whenever k is large enough. ij l
that
0
0
0
1/
u
Let us now modify the sequen e { k } k òN by putting
u lk (x) # ®
I(l)
J(l)
x ò \ U i#1 U j#1 N ijk ; k if x ò N : ij
u k (x) s kj
if
This pro edure is illustrated on Figure 3.10 whi h ounts
m # n # 1,
R
R Ê R & {
},
% ; "}
and the limit energy supported on both omponents of the remainder { spatially homogeneous with the densities equal to
repla ements
a p b:
u lk
uk
ak
S
1/
p
b/k
1
0
"ak Fig. 3.10:
A modi ation of a sequen e {
0
1/
1/l
p
u k }kòN sending a pres ribed energy to the reminder &.
Now we are to prove that the modied sequen e {
l in the u lk }kòN L p ( ; Rm ) attains
sense that
lim X v( u lk (x))g(x) dx k Ù
for any
#X
,
R
Rm
g(x)v 0 (s) l (dxds)
g ò C( ) and v(s) # v0 (s)(1%sp ) with v0 ò R. We have
158
Ë
3 Young Measures and Their Generalizations
lim X v( u lk (x))g(x) dx k Ù
# lim
k Ù
X Il Jl
\Ui#1 Uj#1 N ijk ( )
( )
g(x)v0 (u k (x))(1%u k (x)p ) dx I(l) J(l)
% H H X g(x)v (s kj )(1%s kj p ) dx 0
i #1 j #1 k N ij
#X
X
I(l) J(l)
v0 (s) x (ds)g(x) (dx) % H H v 0 (s j )g(x ij )a lij
Rm
i #1 j #1
N ijk # a lij /s kj p , whi h is implied by the fa t that the volume of the ball of the n n radius r in the spa e R is given by the formula B (1) r and therefore be ause
lim N ik j (1%s kj p ) # a lij ;
k Ù
;
and be ause48
lim X k Ù N ijk
g(x)v0 (u k (x))(1%u k (x)p ) dx # 0:
(3.67)
Ê ( ; ) for l Ù
l Now we want to show that approa hes for any
g ò C( ) and any v0 ò R , we have
lim X l Ù ,
m RR
Indeed, denoting by and
g(x)v 0 (s) l (dxds) # X
X R
Rm
v 0 (s) x (ds)g(x) (dx):
M v 0 ; M g : R% Ù R% respe tively the moduli of ontinuity49 of v 0
g and using (3.61a) and (3.65)(3.66), this onvergen e follows from:
!! !!X !! ! ,
m RR
!!
g(x)v 0 (s) l (dxds) " X
X
!! I ( l ) J ( l ) !! H H v ( s j ) g ( x ) a l 0 ij ij !! ! i #1 j #1
"X
# ¢
48
in the sense that,
m RR
I(l) J(l)
H H X X v 0 ( s j ) g ( x ij ) j i l i #1 j #1 l
To show (3.67), one is to realize that {
X
v 0 (s) x (ds)g(x) (dx)!!!! ! !!
m m R R \R
v 0 (s) x (ds)g(x) (dx)!!!! !
" v (s)g(x) x (ds) (dx) 0
u k p ; k ò
N
} is relatively weakly ompa t in
L1 ( ), hen e also
equi ontinuous due to the DunfordPettis theorem 1.28(iii). Moreover, it is only a simple observation
# limkÙ a lij /s kj p # 0: Let us take " ¡ 0. Due to the equi ontinuity we an nd k ò N su h that, for any k £ k , we have PN k (1%u k (x)p ) dx " g "C0 v "C0 Rm and then we have
that
limkÙ
N ijk
0
the following estimate
0
P k N ij
1
ij
g(x)v0 (u k (x))(1%u k
(
p ( x ) ) d x
¢ g C0 v (
)
)
0
0
1
(
)
%u k (x)p ) dx
C0 (Rm ) PN k (1 ij
" ; whi h proves (3.67). This means that v 0 ( s 1 ) " v 0 ( s 2 ) ¢ M v ( ( s 1 ; s 2 )) and g ( x 1 ) " g ( x 2 ) ¢ M g ( x 1 " x 2 ), where 0 (; ) denotes some metri indu ing the (metrizable) ompa t topology of R m and, of ourse, lim"Ù0% M v 0 (") # lim"Ù0% M g (") # 0 be ause v 0 and g are uniformly ontinuous.
49
R
3.2 Various generalizations
¢
I(l) J(l)
H H X X M v 0 j i l i #1 j #1 l
¢ M v 0
1 l
g
C ( )
1 l
% Mg
g C 0 ( i ) l
% Mg
1 l
1 l
Ë 159
v 0 0 j x (d s ) (d x ) C ( l)
v 0 C 0 (Rm ) ( )
Ù 0
l Ù ):
(for
Now we are in the situation that
lim lim X v( u lk (x))g(x) dx l Ù k Ù
#X
X R
Rm
v 0 (s) x (ds)g(x)(dx)
g ò C( ) and v(s) # v0 (s)(1%sp ) with v0 ò R . By a suitable diagonalization l p m pro edure, one an hoose the net { u } ò L ( ; R ) su h that k for any
l
lim X v( u k (x))g(x) dx # X ò
As the whole net {
l
u k } ò
% P P {
l
R
v 0 (s) x (ds)g(x)(dx) :
Rm
L p ( ; Rm )
and both
C( )
and
R
are sep
N dire ted by the standard ordering. The men
#
u k } ò
m m x (d s ) (d x ) R R \R p m ). in L ( ;
u k }kòN
R
is boundedin
arable, we an even suppose tioned boundedness of {
X
follows from the estimate
¢ C p % P P R Rm
p
u lk L p
m x (d s ) (d x ) \R
(
;Rm )
¢ u k pLp Rm (
;
)
%, where C bounds Å
Remark 3.29 (Embedding of Yp ( ; Rm ) into DiPernaMajda measures). Having ò Yp ( ; Rm ), we an dene an absolutely ontinuous ò r a( ) by the density d ò L ( ) given by p (1% s ) x (d s ) d (x) # 1 % X sp x (ds) and x (ds) # (3.68)
1
d (x)
Rm
Rm while vanishes on R Rm Rm . It is easy to see that ; satises (3.61), so p m thanks to Proposition 3.28. Obviously, (3.60) is also satised that ; ò DM R ; R p m ò DM so that ; is p non on entrating by Lemma 3.27. The measure R ; R m is then given on , R by the formula
orresponding to ; on
\
(
)
(
(
(
)
)
)
(
(
)
)
(d x d s ) # (1% s p )
,(
x (d s ) d x ;
Rm Rm . Besides, Ê
(3.69)
; ) give the same result as when tested by fun tions with the growth stri tly less than p in the sense m that, for any v ò C p (R ) and g ò C ( ), it holds
while it vanishes on the remainder
R
\
)
(
X X
Rm
v(s)
x (d s ) g ( x ) d x
# #
X
,
X X
R
Rm
m RR
g(x)v(s) (d x d s ) 1%sp v(s) 1%sp
x (d s ) g ( x ) (d x ) :
In our last theorem we will show that every DiPernaMajda measure (
(3.70)
; ) whi h does
not satisfy (3.60) an be modied to a DiPernaMajda measure ( ; ) satisfying (3.60)
Ë
160
3 Young Measures and Their Generalizations
; ) and ( ; ) give the same results when tested by fun tions with the growth less than p in the sense that and, simultaneously, both (
X X
for any sure
v(s)
p m R R 1% s
x (d s ) g ( x ) (d x )
#X
v(s)
X
p m R R 1% s
x (d s ) g ( x )
(dx):
(3.71)
v ò C p (Rm ) and g ò C( ). It is natural to address su h DiPernaMajda meaa p non on entrating modi ation of ( ; ). The following assertion es
( ; )
tablishes an expli it formula for it.
Proposition 3.30 (Non on entrating modi ation Kruºík [457℄). Let the ring R be p m separable and let ( ; ) ò DMR ( ; R ) be given. Furthermore, let us dene an abso lutely ontinuous òr a( ) by means of the density d ò L ( ) given by
1
d (x) # X
Rm
and
ò Lw ( ; ; r a( *
R
Rm
))
x (d s )
1%sp
x (d s )
#
Rm
)
Rm
x (d s )
x Rm (d s ) X
Rm
; ) ò DM p ( ; R
X
(3.72a)
by
Then (
"1
is a
x (d s )
:
(3.72b)
pnon on entrating modi ation of ( ; ).
Proof. Let { u k } k òN be a generating sequen e of ( ; ) bounded in
L p ( ; Rm ).
On
R ò ò Yp ( ; Rm ) denes a pnon on entrating DiPernaMajda measure, p m let us denote it by ( ; ) ò DM ( ; R ), whi h satises (3.70) with ( ; ) in pla e of R Yp ( ;
the other hand, this sequen e generates
m ) satisfying (3.70). Due to Re
mark 3.29 this
; ). Altogether we an see that (3.71) is fullled. The formula (3.72a) is just (3.61d) for # whi h was veried in the proof of Proposition 3.28. Then (3.72b) an be obtained "1 p by putting x from (3.64) into the formula x # d ( x )(1% s ) x (d s ); f. (3.68).
(
As the set
p m DMR ( ; R ) is onvex and lo ally ompa t ( f. Example 3.70 below),
its extreme points and rays are of importan e (see Klee's theorem 1.15). We an see that the rays are intimately onne ted with on entration ee ts. Let us still introdu e the shorthand notation
rem( ,
R
Rm # )
% ( ,
òr a
R
Rm ; supp , )
(
)
(
R
Rm Rm \
)
:
Ë 161
3.2 Various generalizations
p m Proposition 3.31 (Geometri al properties of DMR ( ; R ); mostly by Kruºík [458℄). p m òDM ( ; R ) is an extreme point if and only if # i ( u ) for some u ò L p ( ; R m ). (i) R p m In parti ular, extreme DiPernaMajda measures in DMR ( ; R ) must be p non on
entrating.
p p m % t ; t ¡ 0} with some òDM ( ; R m ) ( ; R ) has the form { DMR R m òrem( , R R ). In parti ular, there is no ray in the set of p non on entraand p p m m ting DiPernaMajda measures from DM ( ; R ). Also, any ray in DM ( ; R ) is R R
(ii) Any ray in
0
0
pnon on entrating
omposed from DiPernaMajda measures that have the same modi ation.
p m ( ; R ). DMR p # i ( u ) for some % t ; t ¡ 0} in DM ( ; R m ) is extreme if and only if A ray { R p m m m u ò L ( ; R ) and # Æ x s for some x ò and s ò R R \ R .
(iii) There is no straight line in (iv)
0
0
(
; )
Proof. (Kruºík [458℄.) First, let us realize that an extreme point an be found only
ò DMR ( ; Rm ) not pnon #
on entrating we an write % with # ,Rm and ò rem( , R Rm ) lies on the ray { % t ; t ¡ 0} and therefore it nonvanishing. It follows that su h among
pnon on entrating
p
measures. Namely, for any
annot be an extreme point.
is pnon on entrating but not an extreme point in DMR ( ; Rm ), then 1, 2 ò not an extreme point in the p non on entrating measures, i.e. there exist p m 1 #Ö 2 su h that # ( 1 % 2 )/2. The ( ; R ), both p non on entrating, DMR p
If is
onverse impli ation is trivial.
Û : Yp ( ; Rm ) Û { ò pnon on entrating} ( f. (3.69)), the extreme points in Yp ( ; Rm )
Sin e there is a onetoone ane mapping
p DMR ( ;
R
m );
pnon on entrating DiPernaMajda
are thus mapped uniquely onto extreme points in measures, hen e onto extreme points in in
p m ( ; R ). However, the extreme points DMR
Yp ( ; Rm ) has been already des ribed by Proposition 3.24(i), whi h proves (i). p m By Proposition 3.24(ii), Y ( ; R ) does not ontain a ray and therefore also p non
on entrating DiPernaMajda measures annot ontain a ray.
R
p 2 %(1" t ) 1 ; 2 òDMR ( ; m ) and suppose that (t) # t p m i # % and belongs to DM ( ; ) for any t ¡ 0. Let us make the de omposition i i R ( t ) are p non on entrating and ; ( t ) ò rem( , (t) % ( t ) where ; (t) # i i m ), i # 1 ; 2. It implies that (t) # t 2 % (1" t ) 1 and (t) # t 2 % (1" t ) 1 for any R To prove (ii), let us take 1
R
R
t ¡ 0. We saw that there in no ray in pnon on entrating DiPernaMajda measures, so 1 # 2 . Thus we obtain that (t) # 1 % t 2 % (1" t ) 1 # 1 % t( 2 " 1 ). Putting that # 2 " 1 and 0 # 1 , we get the desired result. p 0 % t ; t ò R} be a line in DM ( ; R m ). To prove (iii), let us suppose that L # { R p 0 % t ; t ¡ 0} and { 0 % t ; t 0} would be rays in DM ( ; R m ), whi h Then both { R m and " belong to rem( , implies by the point (ii) that both R R ). This is possible vanishes, so that L is a singleton. only if
Ë
162
3 Young Measures and Their Generalizations
Let us go on to the point (iv). The end point 0 of the ray in question belongs
p m DMR ( ; R ) and therefore50 the extreme ray must rise from an extreme point of p m DMR ( ; R ). By the point (i), is of the form i(u) for some u ò L p ( ; Rm ). p % t ; t ¡ 0} in DM ( ; R m ) is extreme if and only Due to the denition, a ray { R p ; òDM ( ; R m ): if the following impli ation holds for R to
0
0
1
2
;t ò R% ;r ò (0; 1) su h that % t # r % (1"r ) âá :r ò (0; 1) ;t ò R% su h that % t # r % (1"r) :
0
0
0
0
0
1
0
2
0
1
2
m 1 % 1 and 2 # 2 % 2 with 1 ; 2 òrem( , # R R ). Using this 1 # 2 # 0 (note that representation, we obtain from the above impli ation used for m) 0 # 0 ) that it holds for 1 ; 2 òrem( , R R
We an write 1
;t ò R% ;r ò (0; 1) su h that t # r % (1"r ) âá :r ò (0; 1) ;t ò R% su h that t # r % (1"r) :
0
0
0
0
0
1
2
1
2
t ; t ¡ 0} is an extreme ray in rem( ,
The last impli ation says pre isely that {
R
Rm , )
Å
whi h is possible if and only if is the Dira measure.51
Remark 3.32 (More detailed representations).
In parti ular ases, a more detailed
representation of DiPernaMajda measures an be established. In ase
R
p # 1 u
and
from (3.45), Alibert and Bou hitté [15, Thm. 2.3℄ proved that any sequen e { k } k òN
L1 ( ; Rm ) ontains a subsequen e (denoted, for simpli ity, by the same 1 m m "1 ) , r a% ( ), one has indi es) su h that, for some ( ; ; ) ò Y ( ; R ) , Y( ; ; S bounded in
h u k Ù h Ǳ % (h2 Ǳ
) 
weakly* in
r a( )
h ò Car1 ( ; Rm ) su h that h(x ; s)/(1%s) admits a ontinuous extension on
, R Rm and h2 : , S m"1 Ù R is dened by h2 (x ; s) :# limtÙ s"1 h(x ; ts); note that for h # g v we get just h 2 # g v 2 with v 2 from (3.45). One an verify 1 m that, for a DiPernaMajda measure ( ; ) ò DMR ( ; R ), the AlibertBou hitté rep"1 from (3.54), resentation ( ; ; ) is given by x ( s ) :# a ( x ) x ( s ) where s ÜÙ s m "1 Ù m m denotes the natural homeomorphism S R R \ R , and :# a , where a(x) :# P Rm Rm x (ds); note that is dened a.e. By Lemma 3.27(i), ( ; ) is pfor any
R
\
non on entrating if and only if Another representation of
# 0.52 ò DM1R ( ; Rm )
) with ; ; ò Y ( ; R ; S m "1 ) being the Fonse a measure on
Y( ; 1
(
50
with
R
È ; ) ò r a% ( ) , to , is suggested by
from (3.45), namely
m ) given again by (3.54) and with
orresponding
(
At this point the reader is referred to Köthe [436, Se t. 25℄.
51
Cf. Köthe [436, Se t. 25℄.
52
This was also proved dire tly by Alibert and Bou hitté [15, Thm. 2.6℄. On the other hand, no attain
ability of su h pairs (
; ) has been studied in [15℄.
3.2 Various generalizations
Ë 163
the formula (3.112) bellow. Su h representation has been proposed by Fonse a, Müller and Pedregal [324℄.
p ò [1; %), a similar de omposition, namely ( 1 ; 2 ; ) ò , Y( ; ; S m"1 ) , r a% ( ) with 1x ò r a% (Rm ), has been proposed
For a general
Lw ( ; r a(Rm ))
*
by DiPerna and Majda [266, Thm. 1℄ who showed the onvergen e
h u k Ù (1%0 )h1 Ǳ
1
% (h Ǳ
2
2
) 
weakly* in
r a( )
h ò Carp ( ; Rm ) in the form h(x ; s) # h1 (x ; s)(1%sp ) % h2 (x ; s/s)sp h1 (x ; ) ò C0 (Rm ), where 0 ò L1 ( ) denotes the absolutely ontinuous part of .
for any
with
Remark 3.33 (Testing dis ontinuously a
ording A. Kaªamajska [411, 412℄). A generam Ù R only pie ewise ontinuous with apriori lization for fun tions h with h ( x ; ) : R xed hypersurfa es of possible dis ontinuities has been proposed in [411, 412℄, assuming some nite partition of taken a separable ring
Rm on some open subdomains ( alled bri ks) and then
R of fun tions whose restri tions on these bri ks is ontinuous.
Thus extended DiPernaMajda measures are supported on ompa ti ations of ea h bri ks separately. For further investigation of the set of all DiPernaMajda measures we refer to Examples 3.47 and 3.70 below.
3.2.d
Fonse a's extension of L1 spa es
It is worth ompleting the previous generalizations by a similar onstru tion by Fonse a [316℄ who developed an extension of
L1 ( ; Rm ) whi h an handle positively ho
mogeneous integrands. As su h integrands form a separable (in a natural topology) linear subspa e, we an work in terms of sequen es.
Theorem 3.34 (I. Fonse a [316℄, here modied). Let {u k }kòN be a bounded sequen e in L ( ; Rm ). Then there is a subsequen e, denoted again by {u k }kòN for simpli ity, a mea% m " ) (where S m " denotes again the sure òr a ( ), and a Young measure ò Y( ; ; S 1
1
unit sphere in
R
1
m ) su h that
lim X h(x ; u k (x)) dx k Ù
#X
X
S m"1
h(x ; s) x (ds)(dx) ;
(3.73)
h ò C0 ( , Rm ) su h that h(x ; s) # 0 for x ò :# bd( ) and h(x ; as) # ah(x ; s) m % for any ( x ; s ; a ) ò , R , R . for any
Sket h of the proof. Let us dene the embedding
i ( u ) ; h 0
#
X h 0 x ;
i : L1 ( ; Rm ) Ù r a( , S m"1 ) by
u(x) u ( x )d x u(x)
(3.74)
164
Ë
3 Young Measures and Their Generalizations
h0 ò C0 ( , S m"1 ), i.e. the set of ontinuous fun tions vanishing on , i(u) is a positive Radon measure with the variation equal to P i ( u )( d x d s ) # u L1 ;Rm . Having a bounded sequen e {u k }kòN in L1 ( ; Rm ),
, S m"1 m "1 ) is bounded as well, so that we an sele t a weakly* its image via i in r a( , S for any
S m"1 . Note that
(
)
onvergent subsequen e:
i(u k ) Ù
weakly* in
r a( , S m" ) : 1
(3.75)
Let us denote53
F( ; Rm ) # òr a( , S m" ); 1
;{u k }kòN L ( ; Rm ) : (3:75) holds : 1
Let us dene the mapping
T : C(S m"1 ) Ù r a( ) Ê C0 ( )
*
T v0 ; g
# ; g v
0
(3.76)
by
(3.77)
v0 ò C(S m"1 ) and g ò C0 ( ). Likewise (3.49), we an estimate here < T v0 ; g> ¢ v 0 C S m"1 P g ( x ) (d x ), where # w* lim k Ù u k in r a( ); note that this limit
m "1 ). does exist thanks to (3.75) tested by the fun tions of the form g 1 ò C 0 ( , S Analogously as in the proof of Theorem 3.25, we an dene Tv 0 ò L ( ; ) and estabm "1 ) su h that lish a Young measure ò Y( ; ; S for any (
)
X X
S m"1
; g v 0 # X g ( x )[ Tv 0 ℄( x ) (d x ); g(x)v0 (s) x (ds)(dx) #
(3.78)
vanishes on a set of positive Lebesgue C0 ( ) C(S m"1 ) m "1 ), (3.78) also implies P P ; h 0 > for any h 0 ò in C 0 ( , S h (x ; s) x (ds)(dx) # 0 for s # 0 ; F
f. also (3.50) and realize that it holds also if
measure, whi h an a tually happen here. By the density argument of
we an write, when using (3.74), (3.75), and the identity
lim X h(x ; u k (x)) dx k Ù
# lim
X h 0 x ; k Ù
# ; h As su h
h
0
h ,S m"1 # h0 ,
u k (x) u k ( x )d x # lim i ( u k ) ; h 0 u k (x) kÙ
#X
h0 (x ; s) x (ds)(dx) X
S m"1
#X
h(x ; s) x (ds)(dx): X
S m"1
an range all ontinuous positively homogeneous integrands, (3.73) has
Å
been proved.
F( ; Rm )
; ) ò r a( ) , Lw ( ; ; S m"1 ) 1 m generated in the sense of (3.73) by some sequen e in L ( ; R ); the elements of both F( ; Rm ) and F( ; Rm ) will be addressed as Fonse a measures. Let us denote by
53
Note that any sequen e {
u k }kòN
the set of all pairs (
satisfying (3.75) must be bounded in
*
L1 ( ;
Rm
).
3.2 Various generalizations
Remark 3.35 (Chara terization of Fonse a measures).
Ë 165
We have the simple omplete
hara terization:54
F( ; Rm ) #
( ; );
òr a% ( );
Also note that Theorem 3.34 determines only
ò Y( ; ; S m "1 ) :
a.e.
(3.79)
so that it is arbitrary on
\
supp() whi h may be nonempty; this is a dieren e from the DiPernaMajda measures where always supp( ) # .
Remark 3.36 (Non on entrating Fonse a measures).
Likewise we did for DiPerna
Majda measures, we an dene also here the notion of
; ), whi h will
1non on entrating Fonse a
u ; ) in the sense (3.73), su h that the set {u k ; k ò N} is relatively weakly ompa t in L1 ( ). Here one an show that ( ; ) ò F( ; Rm ) 1non on entrating means pre isely absolutely ontinuous with respe t to the Lebesgue measure. However, ontrary to
measure (
indi ate the existen e of a sequen e { k } k òN , generating
(
DiPernaMajda measures, even sequen es whose energy is not relatively weakly om
1non on entrating Fonse a measure. Indeed, the sequen e from Figure 3.8 (with p # 1) generates the homogeneous Fonse a measure " dx and # Æ whi h an equally be generated by a on( ; ) given by (d x ) # ab x " " stant sequen e u k # ab or also by an sequen e os illating between 0 and 2 ab pa t in
L1 ( )
an onverge to a
1
1
1
1
with the ratio 1:1, et . Thus we an see that the Fonse a measures an re ord mu h lesser information than the DiPernaMajda measures, indeed.
Remark 3.37 (Properties of F( ; Rm )). In view of Remark 3.35, we an observe that m % m " ). In fa t, the triple F( ; R m ) ; r a( , S m " ) ; i with simply F( ; R ) # r a ( , S i from (3.74) forms a onvex  ompa ti ation of L ( ); of ourse, r a( , S m" ) is m
onsidered in the weak* topology. Obviously, F( ; R ) is also lo ally (sequentially) 1
1
1
1
ompa t, and thus it must ontain a ray. It is evident that every ray has just the form
0 ; òF( ; R m ), #Ö 0. Also, F( ; R m ) annot ontain any % t ; t ¡ 0} with some
{0
line. For further investigations of the set of all Fonse a's measures we refer to Examples 3.49 and 3.72.
54
The in lusion
just follows from Fonse a's theorem 3.34. The onverse in lusion in (3.79) must
be proved by a dire t onstru tion: taking a partition a pie ewise homogenization of a given (
k
; ), we
Pk
of
as in the proof of Theorem 3.3 and making k ò r a% ( )
get some pie ewise homogeneous
ò Y( ; S m" ) not uniquely dened, however, be ause we must rst use a suitable extension Ê ( ; ) to make possible testing by only pie ewise ontinuous fun tions. Taking the Young k ò Y ( ; Rm ) dened by k ( s ) # k ( x )"m k ( k ( x )" s ) for s from the sphere of the radius measure x x k (x) and vanishing elsewhere (if k (x) # 0, then kx # Æ ), we an onstru t the sequen e onverging
and of
1
1
to
k
(and also to (
k ;
k
0
) if tested by positively homogeneous integrands) as in the Steps 2b of the
proof of Theorem 3.6. Then passing
k Ù % and making a suitable diagonalization pro edure, we get ; ).
the sought sequen e onverging to (
166
Ë
3.3
A lass of onvex ompa ti ations of balls in
3 Young Measures and Their Generalizations
L p spa es
In Se tions 3.1 and 3.2 we ould see in fa t several on rete onvex ompa ti ations of (bounded subsets in) Lebesgue spa es. The reader might anti ipate that all of them (and also many others) an be overed by a unied way by a suitable general setting. The aim of this se tion is just to onstru t a su iently large lass of onvex ompa ti ations that will in lude the previous ones.
3.3.a
Generalized Young fun tionals YHp % ( ; S) ;
U , the topologi al spa e % ò R% in L p ( ; S) with p ò [1; %℄, i.e.
Let us rst onsider radius
U # B% #
uòL
p
(
to be ompa tied, as the ball of the
; S);
u L p
(
¢ % ;
;S)
(3.80)
where S will again denote a separable Bana h spa e, though mostly we will use merely
S # Rm
in appli ations.
Let us denote by
Carp ( ; S) the linear spa e of all Carathéodory fun tions55 h :
, S Ù R with at most pgrowth, i.e.
p
h(x ; s) ¢
®
a h (x) % b h s S p a h (x) % b h ( s S )
with some
a h ò L1 ( ), b h ò
R% and b h ò C R% (
for p ò [1; %) ; for p # % ;
(3.81)
) nonde reasing. We will onsider
Car ( ; S) as a lo ally onvex spa e endowed by the seminorm  % dened by p
!!
!!
h% # sup !!!!X h(x ; u(x)) dx!!!! : uòB % !
Whenever we will want to emphasize that
(3.82)
!
Car p ( ; S) is endowed with this topology, we
p will write Car% ( ; S) to distinguish it from a ner lo ally onvex topology introdu ed p on
Car ( ; S) later in Se tion 3.4.
Furthermore, we dene the mapping
[
% h℄(u) #
% : Carp ( ; S) Ù C0 (B % ) by
X h ( x ; u ( x )) d x
for h òCarp ( ; S); u ò B % :
(3.83)
Nh : L p ( ; S) Ù L1 ( ) is bounded and ontinuous (see Theorem 1.24), we an see that % h # [ % h ℄() is bounded and ontinuous on p p 0 the ball B % in L ( ; S), hen e % a tually maps Car ( ; S) into C ( B % ). Obviously,
As the Nemytski mapping
55
Re all that the adje tive Carathéodory means that
and
h(x ; ) : S Ù
R
are ontinuous for a.a.
x ò .
h(; s) :
Ù
R
are measurable for all
sòS
3.3 A lass of onvex ompa ti ations of balls in
Lp
spa es
Ë 167
p
h% # % h C0 B % and the onvergen e h Ù h in Car% ( ; S) means pre isely p % h Ù % h in C0 (B % ). Let us note that % is not inje tive, hen e Car% ( ; S) is
(
)
not a Hausdor spa e, f. also (3.7) above. Furthermore, let us dene the embedding
i : L p ( ; S) Ù Carp ( ; S)
*
by
i(u); h For a linear subspa e
#
X h ( x ; u ( x )) d x :
(3.84)
H Carp ( ; S) we dene a linear subspa e FH % C0 (B % ) by ;
FH ; %
# % (H) % onstants on B %
(3.85)
e H : B % Ù FH % as the restri tion e H (u) # e(u)FH % of the evaluation mapping e : B % Ù C0 (B % ) . Moreover, the embedding i H : L p ( ; S) Ù H is dened again by the formula (3.84) for h ò H , i.e. i H ( u ) # i ( u ) H is a restri tion of i on H . The weak* p
losure of i H ( B % ) in H will be denoted by Y H % ( ; S), i.e. *
and
;
;
*
*
*
;
p
YH
;
# ò H ; ;{u } ò B % : #
% ( ; S)
*
Convention 3.38 (Generalized Young fun tionals).
w*lim
ò
i H (u ) :
The elements of
p
YH
;
(3.86)
% ( ; S) will be
alled generalized Young fun tionals. Let us note that for a spe ial hoi e
H # L1 ( ; C(S)), p # % and S S # Rm
the
%, the generalized Young fun tionals oin ides with Young fun tion
ball of the radius
als as stated by Convention (3.1), whi h justies the adje tive generalized. Later we will make the meaning of generalized Young fun tionals even a bit wider; f. Convention 3.65 below.
Theorem 3.39.
Let
H be a linear subspa e of Carp ( ; S), p ò [1; %), and B % and FH
;
%
given respe tively by (3.80) and (3.85). Then: (i)
The linear subspa e
FH ; %
of
C0 (B % )
is onvexifying ( f. Denition 2.4) and thus
FH ; % ); FH ; % ; e H ) is a onvex ompa ti ation of B % . *
(M(
(ii) If
H
is endowed with a topology whi h makes it a lo ally onvex spa e and whi h
is ner than that indu ed from
ompa ti ation of mapping
% .
Carp% ( ; S), then (YHp % ( ; S); H ; i H ) forms a onvex *
;
B % whi h is equivalent with (M(FH
% ) ; F H ; % ; e H ) via the adjoint *
;
*
(iii) If one of the following onditions are satised: 1.
S # Rm , p ò (1; %) and H C( )(Rm )* ontains also the integrand h( x ; s ) # sp ,
or 2.
; M dense in L p ( ; S) :u ò M : h u ò H , where h u (x ; s) # s " u(x) Sp , then the
onvex
ompa ti ation
M(
FH ; % )
is
norm onsistent;
i.e.
iH
is
a
(strong,weak*)homeomorphi al embedding. Proof. To show that {
u } ò
su h that
FH ; %
u1 ; u2 ò B % a net p # %, we an use dire tly
is onvexifying, we must onstru t for any
lim ò e H (u ) # 12 e H (u ) % 12 e H (u 1
2 ).
If
168
Ë
3 Young Measures and Their Generalizations
the onstru tion from the proof of Theorem 3.3 ( f. also Figure 3.3) adopted for the ase of a separable Bana h spa e S instead of
L ( ; S) from L ( ; S) B % .
density of
Rm , while if p ò 1; %
), we an employ56 the
[
p in L ( ; S) in order to approximate
u1 and u2 by some fun tions
Sin e, by the very denition (3.85),
FH ; % always ontains onstants, M(FH ; % ) is a
B % ; f. Proposition 2.5. The point (i) has thus been proved. Let us onsider % restri ted as H Ù F H % . In view of the onsidered topology on H , % is ontinuous and linear. Then a tually % : FH % Ù H . Moreover, % xes B % in the sense that % e H # i H be ause the identity
onvex ompa ti ation of
;
*
*
*
*
;
*
[ %
*
e H ℄(u); h # e H (u); % h # [ % h℄(u) # i H (u); h
h ò H and u ò B % . In parti ular it shows L p norm topology on B % to the weak* topology on H
holds for every
that
*
the
iH
(3.87)
is ontinuous from
e H : B % Ù FH % *
be ause
;
is ontinuous (thanks to the ontinuity of the respe tive Nemytski mappings) and
% , being linear, is also ontinuous in the weak* topologies. Furthermore, we want to prove that the adjoint operator % : F H Ù H is inje tive and has a weakly* ontinuous inverse if restri ted on M # { ò F H % ; < ; 1> # 1}. Indeed, for any 1 ; 2 ò F su h that % 1 # % 2 and < 1 ; 1> # 1 # < 2 ; 1> we H % be ause
*
*
*
*
*
;
*
*
*
;
have the identity
1 ; % h
%
# % ; h % ; 1 # % ; h % ; 1 # ; % h %
*
*
1
1
2
2
2
ò R. Due to the denition (3.85) of FH % , it just means that 1 # 2 . Let us now suppose that we have a net { } ò in M and ò M su h that { % } onverges to % weakly* in H . This implies hòH
valid for any
*
and
;
*
; % h
*
%
# % ; h % ; 1 Ù % ; h % ; 1 # ; % h %
*
*
h ò H and ò R, whi h means pre isely that Ù weakly* in FH . As M(F H % ) M by the very denition of M(F H % ) and % e H # i H by (3.87), we an onp p
lude that M(F H % ) Ê Y H % ( ; S), the ane homeomorphism M(F H % ) Û YH % ( ; S) being just % . *
for any ;
;
;
*
;
;
;
*
Let us go on to the point (iii). In the rst possibility, the weak* onvergen e
i H (u ) means that onverges for any g ò C( ; Rm ) whi h is dense in p L ( ; Rm ) Ê L p ( ; Rm ) and also that , being for h(x ; s) # sp equal to p p m u p L ;Rm , onverges. Sin e 1 p %, the spa e L ( ; R ) is uniformly onvex, and therefore { u } ò must onverge also in the strong topology thanks to the of
*
(
)
FanGli ksberg theorem.
Ù i H (u) means, in par# Ù # u " u pLp S for any u ò M . i
u
For the se ond possibility, the weak* onvergen e H ( )
u "
ti ular, that
56
p u L p ( ;S)
For details we refer to Warga [791, Thm. I.5.18℄.
(
;
)
Lp
3.3 A lass of onvex ompa ti ations of balls in
Ë 169
spa es
u " u L p S ¢ u " u L p S % u " u L p S ¢ 2 u " u L p S % " for ò large enough (depending on u and "). As we may take u p arbitrarily lose to u and " Ù 0, we get eventually u Ù u in L ( ; S). Å
Moreover, one an ertainly estimate ;
(
(
;
)
;
(
)
)
(
;
)
H , we have a large freedom in the hoi e of the topolH . Let us noti e that, by Theorem 3.39, taking a ner p topology on a given H an enlarge only H but not Y H % ( ; S) H . Quite typi ally, parti ular H will be endowed by some norm  H stronger than the seminorm from (3.82), % that means h % ¢ C h H for all h ò H and some C ò R xed. Examples of su h norms We have seen that, for a given
ogy of the linear topologi al spa e
*
*
;
will be given later, f. (3.98) or (3.108). The following assertion points out the topology
Carp% ( ; S) as the a tually limit topology at least if one onsiders the " subspa es H ontaining % ({ onstants on B % }) # { g 1 ò L ( ; C (S))}.57 indu ed from
1
Theorem 3.40.
1
0
H be a linear subspa e of Carp ( ; S) endowed with a lo ally onvex p topology ner than that indu ed from Car% ( ; S), p ò [1 ; %), and F H % be given by p (3.85). Moreover, let H be the losure of H in Car% ( ; S). Then: p p (i) Y H % ( ; S) Ê YH % ( ; S). Let
;
;
(ii) If
;
H
Carp ( ; S) endowed with a topology ner than H H {g 1 ò L ( ; C (S))} but H #Ö H , then
is another linear subspa e of
p that indu ed from Car% ( ; S) and p p Yp ( ; S) ± YH ; % ( ; S). H;%
Proof. Let us onsider the in lusion
1
0
H H
Q : H Ù H . This H and H . Therefore, the
as a linear operator
operator is ontinuous thanks to the hosen topologies on
H into H . Let us show that Q i H # i H . Indeed, we have the obvious identity
adjoint operator
Q
*
maps
*
*
*
Q
*
i H (u); h # i H (u); Qh # % h(u) #
h ò H and u ò B % . p p to prove that Y H % ( ; S) Ê YH % ( ; S).
i H (u); h
valid for every We want
p YH ; % ( ; S)
;
;
By Theorem 3.39(ii) we have
Ê M(FH % ). Thanks to the proper topology of Carp% ( ; S), FH % is just the losure of F H % in C ( B % ), and therefore by Theorem 2.8(ii) we obtain M(F H % ) Ê M(F H % ). p Using again Theorem 3.39(ii) we get eventually M(F H % ) Ê Y ( ; S). This proved the H % ;
;
0
;
;
;
;
;
point (i).
FH % p #Ö FH % . The losedness of H in Car% ( ; S) means pre isely the losedness
Let us go on to (ii). By the assumptions and denition (3.85), we have F p H;% but
Fp H;%
;
;
S#
Rm
, one an show as before that % h is onstant u ò B % ; :a.a. x ò : u(x) S ¢ r} if and only if [ % h℄(x ; ) is onstant on the ball B r S for a.a. x ò . Then passing r Ù , we get that % h is onstant on B % if and only if [ % h℄(x ; ) is onstant on S. If S is not lo ally ompa t, the maximum and the minimum of h ( x ; ) need not be attained but we
an work with " a
ura y, as well. If p # , it su es to take r # % .
57
This an be shown similarly as (3.7): In ase
on {
170 of
Ë
3 Young Measures and Their Generalizations
FH ; % in C0 (B % ). Then M(Fp H;%) ±
get
p ( ; S) H;%
FH ; % ) by Theorem 2.8(iii). By Theorem 3.39, we
M(
± YHp % ( ; S) Ê YHp % ( ; S).
Yp
Å
;
;
The following assertion shows that the lass of onvex ompa ti ations built up by the above pro edure is fairly ri h although, as we will see later in Theorem 3.42, it still does not ontain all onvex ompa ti ations of
Theorem 3.41.
B% . N
p ò [1; %). Then there exist at least 22 dierent onvex ompa tip ations of B % in the form M(F H % ) with H being a linear subspa e of Car ( ; S). Let
;
S is home N, the ardinality of S is surely at least58 the ardinality of N, whi h N 0 2 is known59 as 2 . Then we an take s ; s 0 ò S \ S and put R ( s ) # { v 0 ò C (S); ; v 0 ò 0 C( S) : v0 # v 0 S & v 0 (s) # v 0 (s0 )}; then R (s) is a omplete subring of C (S). In other words, the ompa ti ation R s S is reated from S if the points s and s 0 are glued N 2 dierent manners, whi h gives to ea h other. We an hoose s ò S \ S at least by 2 N 2 at least 2 dierent ompa ti ations R s S of S. Proof. As S ertainly ontains a dis rete ountable subset whose losure in
omorphi with
( )
( )
Then we put
H(s) # C( ) Ôp (R (s)) ;
(3.88)
where Ô
p
:C
0
Rm Ù C p Rm
(
)
(
) is dened by
[Ô
p
v℄(s) # v(s)(1% sp ) :
(3.89)
s1 ; s2 ò S, s1 #Ö s2 implies M(FH s1 % ) ÊÖ M(FH s2 % ). the set N , {nite subsets of C0 (S)} dire ted by the ordering ¢ , . It is lear that, for any # ( k ; { v 1 ; ::: v L }) ò , the set N ( s j ) # { s ò S; max1¢l¢L v l (s) " v l (s j ) ¢ 1/k} forms a neighbourhood of s j in S, j # 0; 1, with v l standing for the ontinuous extension of v l . Therefore, for any ò , we an nd s j ò N (s j ) and then, obviously, the net {s j } ò onverges to s j in S. We an even suppose s j ò S be ause S is dense in S. As s j ò S \ S, the set B % # { s ò S; s £ % } is also a neighbourhood of s j so that lim ò s j # %. The image of s 1 via the anoni al surje tion S Ù R s 1 S is glued with the image of s 0 while R s 2 S glues only s 2 with s 0 so that the image of s 1 via the anoni al surje tion S Ù R s 2 S remains separated from the image of s 0 provided s 1 #Ö s 2 , as supposed. This means that lim ò [ v ( s 1 ) " v ( s 0 )℄ # 0 for every v ò R( s 1 ) while lim ò [v(s1 ) " v(s0 )℄ #Ö 0 for some v ò R(s2 ).
We want to show that, for any
(
);
(
);
Let us take as the index set
;
;
;
;
(
(
(
)
;
;
58
)
)
;
;
More pre isely, it is even equal to the ardinality of
N
if
S is separable, as supposed; f. Engelking
[284, Thm. 3.5.3℄.
59
See Bourbaki [144, Chap.IX, Exer ise 1.12℄ or Engelking [284, Corollary 3.6.12℄ for details.
Lp
3.3 A lass of onvex ompa ti ations of balls in
Choose some x 0 ò and, for any ò , a neighbourhood N j N j # % p /(1%s j p ) and, denoting j # N j , let us put ;
;
;
;
u j (x) #
sj
;
0
of
x0
su h that
;
if
;
Ë 171
spa es
if
x ò j ; x ò \ j ; ;
;
j # 0; 1. Note that u j L p ;S # %s j /(1%s j p )1 p % so that ea h u j belongs to B % . As lim ò j # 0, we may also suppose lim ò diam( j ) # 0. p n Let us take some h # g Ô v with g ò C ( ) and v ò R( s 1 ). As R is ompa t, g % % is uniformly ontinuous on so that there is the modulus of ontinuity M g : R Ù R of g , i.e. lim " Ù0 M g ( " ) # 0 and g ( x 1 ) " g ( x 2 ) ¢ M g ( x 1 " x 2 ) for any x 1 ; x 2 ò . Then for
;
(
)
;
/
;
;
;
;
!! ! !! e H ( s ) ( u 1 ; ) " e H ( s ) ( u 0 ; ) ; % h !!! 1 1 !! !! # !!!!X g(x)1%s1; p v(s1; ) dx " X g(x)1%s0; p v(s0; ) dx!!!! ! 1 !
0 !! !! ¢ !!!!X g(x0 )1%s1; p v(s1; ) dx " X g(x0 )1%s0; p v(s0; ) dx!!!! ! 1 !
0 ;
;
;
;
%
H X g(x) j #0 ; 1 j
" g(x
1% s j ;
0 )
p !! )! v ( s j ; ) d x !
;
¢ % p !!!!g(x
! v ( s 1 ; )" v ( s 0 ; )!!!
0)
onverges to zero if
ranges
%
H % j #0 ; 1
p
M g (diam( j ))v(s j )
;
the same luster points in M(F H ( s 1 ); % ). On the other hand, taking
h#1
e H(s
2)
(
v ò R(s2 )
su h that
Ôp v, we obtain obviously
u1 ) " e H ;
(
;
. For a general h ò H(s1 ) we an obtain the same result e u1 )} ò and {e H s1 (u0 )} ò must have
analogously. This shows that the nets { H ( s 1 ) (
putting
;
s2) (u0; ); % h
#
(
lim ò [v(s
1;
)
)
;
" v(s
0;
)℄
#Ö 0 and
p X 1% s 1 ; v ( s 1 ; ) d x
1 ;
"X
p 1% s 0 ; v ( s 0 ; ) d x
0
# % p v(s
1;
)
" v(s
0;
)
;
;
whi h does not approa h zero. In other words, the luster points of the nets {
eH
(
s 2 ) ( u 1 ; )} ò and { e H ( s 2 ) ( u 0 ; )} ò an be separated in M(F H ( s 2 ); % ). This shows that
denitely M(F H ( s 1 ); % )
ÊÖ M(FH s2 (
);
Å
% ).
Let us omplete the properties of the ordering of the lass of onvex ompa ti ations of
B%
presented here. Let us emphasize that two onvex ompa ti ations
need not possess a supremum in the lass of all onvex ompa ti ations of a given
B%
( f. Example 2.3), so that the situation stated in the following theorem is rather
ex eptional.
Theorem 3.42 (Properties of the ordering). M(F H ; % );
The lass of onvex ompa ti ations
H a linear subspa e of Carp ( ; S)
(3.90)
172
Ë
3 Young Measures and Their Generalizations
of the ball
B%
²
ordered by the relation is a latti e and, for two subspa es
Carp ( ; S), it holds
sup M(FH1 % ); M(FH2 % ) # ;
;
inf M(FH1 % ); M(FH2 % ) # ;
;
H1 ; H2
FH1 %H2 ; % ) ;
M(
FH 1 H 2 ; % ) ;
M(
p where H j denotes the losure of H j in Car % ( ; S). Moreover, the lass (3.90) possesses the nest element, namely M(FCarp ( ;S); % ), whi h is, however, not any maximal onvex
ompa ti ation of
B % in general.
H1 ; H2 linear subspa es of Carp ( ; S), it is obvious that both H 1 H 2 p and H 1 % H 2 are also linear subspa es of Car ( ; S). Both subspa es generate via 0 (3.85) some onvexifying subspa es of C ( B % ); as for H 1 % H 2 , it is essential that p % (Car ( ; S)) itself is a onvexifying subspa e of C0 (B % ). 1 0 Let us put H 0 # { g 1 ò L ( ; C (S))}.60 We will show that M(F H % ) ² M(F H % ) implies H H % H 0 . First, thanks to Theorem 2.8 we have M(F H % ) Ê M(F H % H % ) be0 0
ause F H % H % is the losure in C ( B % ) of F H % . Therefore, M(F H % H % ) ² M(F H % H % ). 0 0 0 Proof. For
;
;
;
;
As both
FH %H0 ; %
and
FH %H0 ; %
;
;
are losed, we an dedu e that
;
FH %H0 ; %
;
FH %H0 % ; f.
;
H % H0 # %"1 (FH %H0 % ) and H % H0 # %"1 (FH %H0 we an also dedu e that H % H 0 H % H 0 , and therefore H H % H 0 , as well.
the proof of Corollary 2.9. As
;
;
% ),
Let us now prove that M(F H 1 % H 2 ; % ) is the supremum (i.e. least upper bound) of M(
FH1 ; % )
³ M(FH j ; % ) for j # 1; 2 FH1 %H2 ; % FH j ; % due to the denition (3.85). that M(F H ; % ) ³ M(F H j ; % ) for j # 1 ; 2 implies M(F H ; % ) ³
and M(F H 2 ; % ). First, it is lear that M(F H 1 % H 2 ; % )
thanks to Theorem 2.8 be ause obviously Se ondly, we have to show
H j H % H . As H % H is a linear spa e, we have got also H % H H % H . This implies M(FH1 %H2 % ) ² M(F H % H % ) Ê M(F H % ) due to Theorem 2.85 be ause F H % H % is the losure in C ( B % ) 0 0 p of F H % thanks to the appropriate topology on Car% ( ; S). M(
FH1 %H2 ; % ).
Indeed, we have already demonstrated that 1
2
0
;
0
;
;
0
0
;
;
Let us now prove that M(F H
1 H 2 ; %
) is the inmum (i.e. greatest lower bound) of
) ² M(F H j ; % ) for j # 1 ; 2 1 H 2 ; % FH 1 H 2 ; % FH j ; % and be ause M(FH j ; % ) Ê 0 M(F H j ; % ) sin e F H ; % is the losure in C ( B % ) of F H j ; % . Se ondly, we have to show that j M(F H ; % ) ² M(F H j ; % ) for j # 1 ; 2 implies M(F H ; % ) ² M(F H H ; % ). Indeed, we have 1 2
FH1 ; % )
M(
and M(F H 2 ; % ). First, it is lear that M(F H
thanks to Theorem 2.8 be ause obviously
H H j % H0 for j # 1; 2, and therefore also H H 1 H 2 % H0 . This implies M(F H % ) ² M(F H H % H % ) Ê M(F H H % ) due to the denition of H 0 and 1 2 0 1 2 already demonstrated that ;
;
;
(3.85).
p ò [1; %) and # 1 , 2 for j some n j ¡ 0, j # 1; 2; n # n1 % n2 . It is known that L p ( ; S) p p is isometri ally isomorphi with L ( 1 ; L ( 2 ; S)) via the mapping T dened by p [ Tu ℄( x 1 ) # u ( x 1 ; ). Let us put S # L ( 2 ; S), whi h is again a separable Bana h Let us now onsider a spe ial ase
domains in
Rn j
with
60
Note that
H0
# %"
1
({ onstants}) and thus it is losed in
Carp% ( ; S) be ause % is ontinuous.
Lp
3.3 A lass of onvex ompa ti ations of balls in
Carp ( ; S) and apply our theory dire tly.
spa e, so that we an speak about the spa e
1
% (Car ( 1 ; S)) is a onvexifying subspa e of C0 (B % ), : Car ( 1 ; S) Ù C (B % ) is dened by [ % h ℄(u) # P h (x1 ; Tu(x1 )) dx1
This implies in parti ular that
p
%
where
Ë 173
spa es
p
0
1
h ò Car ( 1 ; S) and u ò B % . We want to show that the onvex ompa ti % (Carp ( 1 ; S))) of B % is stri tly ner than the onvex ompa ti ation p p M( % (Car ( ; S))) # M(FCarp ;S % ). Indeed, having h ò Car ( ; S) we an always p nd h ò Car ( 1 ; S) su h that % h # % h , namely h ( x 1 ; s ) # P h(x1 ; x2 ; s (x2 )) dx2
2 p for s ranging L ( 2 ; S). Indeed, by the Fubini theorem, for any u ò B % we have for
p
ation M(
(
);
% h (u)
# X h (x ; Tu(x 1
1
1 ))
dx # X 1
X h ( x 1 ; x 2 ; u ( x 1 ; x 2 )) d x 2 d x 1
2
1
# X h(x ; u(x)) dx # % h (u) :
Let us also note that a tually
L p (
2
h ò Carp ( 1 ; S)
be ause the ontinuity of
h (x1 ; ) :
R follows by the standard properties of the Nemytski mappings (see
; S) Ù
Theorem 1.24) and be ause of the estimate
h (x1 ; s ) ¢
¢
X h(x1 ; x2 ;
2
s (x2 ))dx2
X a h (x1 ; x2 )
2
% b h s (x
a h ò L1 ( 1 ) is dened by a h (x1 ) #
where
p S
dx # a h (x ) % b h s Sp ;
2 )
2
1
a h (x1 ; x2 ) dx2
P
2
and
b h # b h
with
ah
and
b h the oe ients from (3.81). This shows that % (Car ( 1 ; S)) % (Car ( ; S)), from % (Carp ( 1 ; S))) ³ M( % (Carp ( ; S))) follows by Theorem 2.8. 1 2 Let us take two sequen es { u } k òN and { u } k òN in B % . The former one has the k k 1 properties that u ( x 1 ; ) is onstant on 2 for a.a. x 1 ò 1 and takes only the values k s ; "s ò S with some s #Ö 0 su iently small, and the whole sequen e {u1k }kòN on2 2 1 p verges weakly in L ( ; S) to 0. The latter sequen e { u } k òN is dened by u ( x ) # u ( x ) k k k 2 1 if x $ ( x 1 ; x 2 ) ò 1 , 2 and u ( x ) # " u ( x ) if x $ ( x 1 ; x 2 ) ò 1 , 2 , where 2 and 2 k k are some disjoint measurable parts of 2 of a positive measure su h that 2 2 # 2 . Let us note that both sequen es are bounded even in L ( ; S), so that we an interp
whi h M(
p
pret them by means of suitable Young measures; f. Example 3.44 or 3.45 below. Then, roughly speaking, both sequen es onverges to the homogeneous Young measure
# 21 Æ s % 12 Æ"s in the representation of M(FCarp S % ). On p the other hand, on the representation of M( % (Car ( ; S))), the rst one onverges to the homogeneous Young measure ò Lw ( ; rba(S)) given by x1 # 12 Æs1 % 21 Æ"s1 with s ( x ) # s , while the se ond one onverges to ò Lw ( ; rba(S)) given by 1 2 1 2 x 1 # 2 Æs % 2 Æ "s with s ( x ) # s if x ò and s ( x ) # " s if x ò . It is lear that p #Ö be ause they an be separated by any h ò Car ( ; S)) with the properties h (x ; s ) # h (x ; " s ) #Ö h (x ; s ) # h (x ; " s ). This shows that M( % (Carp ( ; S))) ± M( % (Carp ( ; S))) so that M( % (Carp ( ; S))) is not a maximal onvex ompa ti ation of B % . Å ò Lw ( ; rba(S)) given by
*
x
(
1
1
1
2
2
1
2
2
2
2
2
1
1
2
2
1
1
1
2
1
*
2
1
2
1
2
2
);
1
1
*
;
1
Ë
174 3.3.b
3 Young Measures and Their Generalizations
The omposition h Ǳ
u ò B % and a Carathéodory integrand h, we spoke about a omposition h u h u℄(x) # h(x ; u(x)). We saw in (3.3) that there is a natural generalization if, instead of u , we onsider a Young measure # { x }xò . Then the omposition h Ǳ results to a fun tion h Ǳ : x ÜÙ PS h(x ; s) x (ds); note that if x # Æ u x , then P h ( x ; s ) x (d s ) # h ( x ; u ( x )) # [ h u ℄( x ) so that the omposition h Ǳ a tually extends S the omposition h u . If the Young measure is onsidered61 as a Young fun tional 1 ò L ( ; C(S)) , the reader an easily verify that this fun tion, denoted by h Ǳ , an be alternatively dened by the identity < h Ǳ ; g > # < ; g  h > for any g ò L ( ), where g  h abbreviates (g 1)  h, i.e. For a given
dened by [
(
)
*
[
g  h℄(x ; s) # g(x)h(x ; s) :
To make possible a study of lo al properties of generalized Young fun tionals, we
an perform this onstru tion even in general situations, the result being a ertain
, however. For a linear subspa e C( ) G L p subspa e H Car ( ; S) is G invariant if
measure on
(
), we will say that the
GH # H;
:g ò G :h ò H : g h ò H . For h ò H G omposition h Ǳ ò G by whi h means
(3.91) and
òH
*
, let us then dene the
*
h Let us note that a tually
h
G
Ǳ ; g # ; g h :
(3.92)
G
h Ǳ i H (u) # Nh (u) # h u be ause, for any g ò G,
G
Ǳ i H ( u ) ; g # i H ( u ) ; g  h #
The dual to the intermediate spa e
G
X g ( x ) h ( x ; u ( x )) d x
# h u ; g :
orresponds to a ertain spa e of measures,
though we do not want to spe ify su h measures in details; f. also Example 1.33. Of ( ), these measures are just r a( ) vba( ), respe tively; f. Theorem 1.32. Yet, these limit ases may bear sometimes
ourse, in the limit ases or
G # C( )
or
G # L
disadvantages as the former one does not allow multipli ation by dis ontinuous fun tions while the latter one reates a nonmetrizable weak* topology on bounded sets in
G
*
. For these reasons, a usage of an intermediate spa e
C ( ) #Ö G #Ö L
(
) may ap
pear a tually advantageous espe ially for development of a numeri alapproximation theory; f. As
G # lG0 from (3.164) below.
G
G will be mostly lear from a ontext or the result g Ǳ h will not depend on G,
Ǳ
G
Ǳ
we will write simply instead of . Let us note that to ensure a tually
61
Cf. (3.4) together with the Convention 3.1.
h Ǳ òG
*
we
3.3 A lass of onvex ompa ti ations of balls in
must require, in addition to (3.91), the ontinuity of the mapping
Lp
spa es
Ë 175
g ÜÙ g  h : G Ù H .
However, to guarantee an a tually good sense of this omposition, we will have to impose even a bit stronger assumption:
:g ò G; h ò H :
g  h H ¢ C g L
(
) h H
:
(3.93)
Proposition 3.43 (Properties of the omposition Ǳ ). spa e of
L
(
Let G C ( ) be a linear sub ) and H be a Ginvariant normed spa e with the norm  H ner than  %
and satisfying (3.93). Then: The bilinear mapping ( h ;
(i)
,
) ÜÙ h Ǳ : H , H Ù G *
*
*
to the weak* topology
G . p p If ò YH % ( ; S) is attainable by a net { i H ( u )} ò su h that the set { u ; ò } is S 1 1 relatively weakly ompa t in L ( ), then h Ǳ ò L ( ) for any h ò H . p If ò YH % ( ; S) is arbitrary but h satises additionally the growth ondition on
(ii)
is jointly ontinuous from the
H,H
(strong weak*)topology on the bounded subsets of *
;
(iii)
;
;a h ò L q ( ) ;b h ò R% :
p/ q
h(x ; s) ¢ a h (x) % b h s S
(3.94)
1 q ¢ %, then h Ǳ ò L q ( ). q If, for 1 q , h ò H satises g  h ò H for any g ò L ( ) and g  h H ¢ C h g L q q with some C h ò R, then h Ǳ ò L ( ) even for any ò H .
for some (iv)
(
)
*
h ò H and ò H , we want to show that: :" ¡ 0 :g ò G :R ò R% ;Æ ¡ 0 ;h0 ò H :h1 ò H :1 ò H , 1 H ¢ R *
Proof. For any
*
*
max " ; h 1
This is a tually true for
0
; h 1
" h H ¢ Æ
âá
h1 Ǳ 1 " h Ǳ ; g ¢ " :
h0 # g  h and 0 Æ ¢ 2"1 " min(1; (CR g L
(
"1 ). Indeed,
) )
we an estimate
h1 Ǳ 1 " h Ǳ ; g ¢
#
h 1 Ǳ 1 " h Ǳ 1 ; g %
1 ; g  (h1 " h) %
h Ǳ 1 " h Ǳ ; g
1 " ; g  h ¢ C 1 H g L h1 " h H " " % 1 " ; h0 ¢ ÆCR g L % Æ ¢ % ¢ " :
*
(
2
)
(
)
2
The point (i) has been thus proved.
u
p { u ; S
B%
lim ò i H (u ) #
H and ò } is relatively weakly ompa t in L ( ). As the mapping Ù Ü h Ǳ was shown to be ontinuous, we an see that lim ò h Ǳ i H ( u ) # h Ǳ . Simultaneously, we p p p have the estimate h Ǳ i H ( u ) # h u ¢ a h % b h u with a h and b h from (3.81), S p 1 whi h shows that the set { h Ǳ i H ( u ) ; ò } is relatively weakly ompa t in L ( ), p as well. Therefore the limit of the net { h Ǳ i H ( u ) } ò , whi h is just h Ǳ , must live in L1 ( ), proving thus (ii). Taking a net { u } ò in B % su h that lim ò i H ( u ) # weakly* in H , the assumpConsider a net { } ò in
su h that
weakly* in
*
1
*
tion (3.94) allows us to estimate
h Ǳ i H (u ) L q
"
(
)
"
# hu L q ¢ a h L q % b h """"" u Sp q """""L q (
)
(
)
/
# a h L q % b h u pL pq S : /
(
)
(
)
(
;
)
Ë
176
3 Young Measures and Their Generalizations
h Ǳ i H (u )} ò is bounded in L q ( ), hen e it must have a weak q q (or weak* if q # %) luster point in L ( ). As L ( ) is naturally embedded into G q ( f. Example 1.34) and this net onverges in G to h Ǳ , this luster point in L ( ) must q
oin ide with h Ǳ , whi h shows that h Ǳ ò L ( ). Thus (iii) was shown. This shows that the net {
*
*
The point (iv) will be shown if one realizes that, thanks to the estimate
h Ǳ ; g
# ; g  h ¢ H g  h H ¢ C h H g L q *
g ÜÙ is q itself must belong to L ( ). the mapping
Let us remark that bounded in
H
*
p
YH
;
*
a ontinuous linear fun tional on
% ( ; S) is always a bounded subset of
H
(
)
;
L q ( )
so that
Å
i
*
hǱ
be ause H (
B % ) is
thanks to the estimate
i H (u) H ¢ sup *
h H ¢1
!! ! !![ % h ℄( u )!!!
¢ C sup !!!![ % h℄(u)!!!! ¢ C
(3.95)
h % ¢1
C is here the onstant from the assumed estimate h% ¢ C h H . Therefore, the h ; ) ÜÙ h Ǳ stated in Proposition 3.43(i) is parti ularly relevant for p p ranging YH % ( ; S). Also let us remark that, supposing ò YH % ( ; S), h Ǳ has a good p sense not only for h ò H but even for h belonging to the losure H of H in Car% ( ; S). p p This is lear by Theorem 3.40, whi h says in parti ular that Y H % ( ; S) Ê YH % ( ; S).
where
joint ontinuity of (
;
;
p
;
;
ÜÙ h Ǳ : YH % ( ; S) Ù G is, in fa t, a onp 1 tinuous ane extension of the Nemytski mapping N h : L ( ; S) Ù L ( ) generated by h ; f. Example 3.96. Let us also remark that the mapping
*
;
We an also generalize the notion homogeneous introdu ed so far only for las
p
ò YH
% ( ; S) hoh Ǳ is onstant in for every h ò H su h that h(; s) is onstant for ea h p s ò S, or equivalently for every h ò H in the form h # 1 v. Analogously, ò YH % ( ; S) will be alled pie ewise homogeneous on a given partition of if h Ǳ is pie ewise
onstant on this partition whenever h ò H is su h that h ( ; s ) is pie ewise onstant for any s ò S. si al Young measures. We will all a generalized Young fun tional
;
mogeneous if
;
3.3.
Some on rete examples
It is time to present some examples viewed from the perspe tive of the theory of generalized Young fun tionals.
Example 3.44 (Classi al Young measures). Let us take p # , S # Rm , and H # Car ( ; Rm ). Let us note that, in fa t, we an equally work with Carp ( ; Rm ) for any p £ 0 be ause Carp ( ; Rm ) ,S % # Car ( ; Rm ) ,S % for S % # {s ò Rm ; s ¢ %} the ball
3.3 A lass of onvex ompa ti ations of balls in
in
Rm of the radius %. Hen e, we will better take H # Car
0
h
norm Car0 ( ;Rm )
spa es
Ë 177
; Rm ) endowed with the
# P supsòRm h(x ; s)dx. This strong topology is a tually ner than Car ( ; Rm ) by the seminorm  % be ause of the estimate 0
the topology indu ed from
(
Lp
h% # % h C0
(
B% )
!! !! !! ! !!X h ( x ; u ( x )) d x !!! ! u ( x )¢ % !
# sup
¢ X sup h(x ; s)dx # h L1 S % ¢ h Car0 Rm : (
s ¢ %
;
)
(
;
(3.96)
)
B% # U S # S % . The set Y H % ( ; Rm ) then ontains just the Young fun tionals m (i.e. Y H % ( ; R ) Ê Y ( ; S % ), whi h justies the notion of generalized Young fun tion0 0 m m als for a general H dierent from Car ( ; R ). Moreover, H # Car ( ; R ) is L ( )Then it is obvious that we have re overed the situation from Se tion 3.1 with
given by (3.1) for
;
;
invariant and (3.93) is satised due to the obvious estimate
g  h H #
X sup g ( x )
s ¢ %
¢ g L
(

h(x ; s)dx #
) sup h ( ; s ) L 1 ( ) s ¢ %
X g ( x ) sup h ( x ; s )d x
s ¢ %
# g L
(
) h H
:
Example 3.45 (Fattorini's generalization). Let us take p # %, S a separable Bana h H # L ( ; C (S)) endowed with the norm h L1 C0 S # P supsòS h(x ; s)dx. This strong topology is again ner than the topology indu ed from Car ( ; S) be ause spa e,
1
0
(
;
(
))
0
the estimate (3.96) applies here as well. Then it is obvious that we have re overed the
B % # U given by (3.1) for S the ball in S of the radius
)invariant, satisfying (3.93). Besides, we ould also take H # Car0 ( ; S) whi h is possibly larger than L1 ( ; C0 (S)). It would reate also a onvex ompa ti ation of B % # U given by (3.1) for S the ball in S. The question whether it is equivalent with the onvex ompa ti ation 1 0
reated by H # L ( ; C (S)) is open, however.
situation from Se tion 3.2.a with
%. Again, su h H is L
(
Example 3.46 (S honbek's generalization: the L p Young measures). p ò [1; %), S # Rm , and H # C( ) C p (Rm )
with
Let
us
take
(3.97)
C p (Rm ) dened by (3.33). We an endow H by the norm
h H #
sup
m ( x ; s )ò ,R
h(x ; s) : 1%sp
This strong topology is ner than the topology indu ed from the estimate
(3.98)
Carp% ( ; Rm ) be ause of
Ë
178
!! !! !! h !!%
3 Young Measures and Their Generalizations
!! ! !!X h ( x ; u ( x )) d x !!! !! !! ! u L p ;R m ¢ % !
# % h C0 B % # (
)
sup (
¢
)
sup
X (1% u ( x )
u L p ;R m ¢ % (
)
)
¢ ( % % p )
sup
m ( x ; s )ò ,R
h(x ; u(x)) dx 1%u(x)p
h(x ; s) # 1%sp
(
% % p ) h H :
(3.99)
H is C( )invariant, satisfying also (3.93). We want to show that the orrespondp m
onvex ompa ti ation Y H % ( ; R ) is equivalent with a ertain sets of Young
Su h ing
p
;
measures, namely
Y% ( ; Rm ) # p
ò Y( ; Rm );
X X s
Rm
p
x (d s ) d x
¢ % :
(3.100)
Y% ( ; Rm ) ontains just those ò Y( ; Rm ) su h that # w* lim ò Æ(u ) for some net {u } ò su h that u L p ( ;Rm ) ¢ %, where m the embedding Æ : B % Ù L w* ( ; r a(R )) is dened again by (3.15). Let us also rem m * mind that r a(R ) Ê C 0 (R ) by the Riesz theorem 1.32(iv) and, by the Dunford m 1 m * Pettis theorem, we an see that L w* ( ; r a(R )) Ê L ( ; C 0 (R )) . Of ourse, the p m m natural topology on Y % ( ; R ) is just the weak* topology of L w* ( ; r a(R )). Let us p 1 m m abbreviate H 0 # L ( ; C 0 (R )). By the very denitions Y % ( ; R ) # w* l Æ ( B % ) and p p p Y H0 ; % ( ; Rm ) # w* l i H0 (B % ), we an see that Y% ( ; Rm ) Ê Y H0 ; % ( ; Rm ) via the mapm * ping : L w* ( ; r a(R )) Ù H dened by ( f. also (3.14)) In view of the proof of Proposition 3.22,
(
)
; h #
p
X X
Rm
h(x ; s)
x (d s ) d x :
R
R
p m ) Ê Y p ( ; m ) provided we ( ; ;% H; % m m) show the losures of H 0 and H ) to be equal to ea h other. As C 0 ( m ) is dense62 in L 1 ( ; C ( m )) in the standard norm of is separable, C ( ) C 0 ( 0 1 m L ( ; C0 ( )), hen e in the seminorm  % , as well. Now we have C( ) C0 ( m ) H , so that we are to show that C( ) C ( m ) is dense in H in the topology of Furthermore, by Theorem 3.40 we an get
R
R
p in Car% ( ;
Y H0
R
R
R
0
R
R
Carp% ( ; Rm ). In other words, it su es to show that every h # g v with g ò C( ) m m and v ò C p (R ) an be approximated by some h " ò C ( ) C (R ) in the seminorm  % . Let us take h " ( x ; s ) # g ( x ) v ( s ) v " ( s ) with the uto fun tion v r ( s ) dened by m % % (3.38). Sin e v ò C p (R ), there is a : R Ù R ontinuous su h that v ( s ) ¢ a ( s ) and limtÙ% a(t)t"p # 0. We an additionally require limtÙ% a(t) # %. Still there is a % %
ontinuous fun tion b : R Ù R su h that lim t Ù% b(t)/t # % and b(a(t)) ¢ Ct p
0
1/
62
We an onsider
Rm
C0 (
) as a losed subspa e of
(Alexandro ) ompa ti ation of Thm. I.5.25℄.
Rm
C(
Rm
) where
Rm
denotes here the onepoint
, whi h is a metrizable ompa t, and then apply Warga [791,
3.3 A lass of onvex ompa ti ations of balls in
Lp
spa es
Ë 179
C ò R% . Therefore, by the DunfordPettis and the de la ValléePoussin theorems 1.28(ii+iv), the set { a ( u ); u ò B % } is uniformly integrable in the sense for some
lim
sup
k Ù% u L p ;Rm (
)
X
¢%
{
x ò ; a ( u ( x ))£ k }
a(u(x)) dx # 0 :
Then we an estimate
h " h " % #
!! !!X g ( x ) v ( u ( x ))(1 !! u L p ;R m ¢ % !
sup (
¢ g C (
¢ g C (
¢ g C (
whi h tends to zero with
"v
1/
)
)
)
)
sup
u L p ;Rm
u L p ;Rm
u L p ;Rm
(
X
)
¢%
)
¢%
)
¢%
sup (
sup (
X {
X {
!! ! " ( u ( x )) d x !!! !
a(u(x))(1 " v1
x ò ; u ( x )£1/ " }
/
" ( u ( x )) d x
a(u(x)) dx
x ò ; a ( u ( x ))£ a (1/ " )}
a(u(x)) dx ;
" Ù 0 be ause a(1/") approa hes %.
Y% ( ; Rm )
Altogether we have thus showed that the set of Young measures
p equivalent with Y H ; % ( ;
H
Moreover,
p Y H ; % ( ;
R
R
m ) for
H from (3.97).
from (3.97) an be enlarged so that
m ) with
H # L
(
Y% ( ; Rm ) p
p
is equivalent also to
) C p (Rm ) ;
(3.101)
whi h an be proved by the same arguments using also the obvious in lusion
C0 (R
m)
L
(
) C0 (R
m)
L ( ; C 1
R
0(
is
m )).
C( )
Moreover, this onvex ompa ti ation is not norm onsistent.63
Example 3.47 (The generalization by DiPerna and Majda).
Rm , R a omplete subring of C Rm , and 0
(
Let us take
p ò [1; %), S #
)
H # C ( ) Ô p (R ) ;
(3.102)
p from (3.89). Again we an endow
H by the norm h H dened by (3.98), whi h p m indu es a ner topology than the topology indu ed from Car ( ; R ) be ause of (3.99). p m Then the respe tive onvex ompa ti ation Y H % ( ; R ) is equivalent with a ertain p m subset of DM ( ; R ) from (3.51), namely the set R
with Ô
;
p m DMR % ( ; R ) # òr a( ,
;
R
Rm ; )
;{u } ò ; u L p Rm ¢ % : (
63
)
w*
lim i(u ) # ò
(3.103)
u k }kòN from Figure 3.7 with ab p # %. Then ¡ 0 so that u k does not onverge to 0 in the L p strong topology but Æ(u k ) Ù Æ(0)
To see this, it su es to take a sequen e {
u k Lp ( ;Rm ) # % p weakly* in Y % ( ;
;
Rm
).
1/
Ë
180
3 Young Measures and Their Generalizations
i # (JR )"1 i p *
where
was dened in the proof of Theorem 3.25. Of ourse, the ane
Rm
p m DMR % ( ; R ) is just the mapping Ô p JR : r a( , R Rm ) Ù H where Ôp : H Ù C( ) R is the isometri al isomorphism h ÜÙ h/(1%sp ). The veri ation that it xes B % , whi h means here Ôp JR i # i H , is
homeomorphism between *
p
YH
;
% ( ;
*
) and
;
*
*
*
an easy exer ise. This onvex ompa ti ation is also norm onsistent.64 Let us emphasize that the onvex ompa ti ation generated by (3.102) is stri tly ner than that one generated by (3.97). To see it, we an take the net indi ated on
i 0 i 0 p for example, the integrand h ( x ; s ) # s . Let us note that H from (3.102) is C ( )invariant, satisfying also (3.93). Yet, it is not Ginvariant for any G C( ), G #Ö C( ). This is a ertain disadvantage of the DiPerna
Figure 3.7, whi h annot be distinguished from H ( ) in the latter onvex ompa ti
ation while it an be separated from H ( ) in the former onvex ompa ti ation by,
Majda measures, whi h auses di ulties espe ially within an approximation theory ( f. Se tion 3.5) and whi h may sometimes make this onvex ompa ti ation insu iently oarse.
Example 3.48 (A renement of DiPernaMajda measures).
To put o the disadvan
tage mentioned in the last example, we are tempted to enlarge a bit the spa e (3.102). Having some ring
G su h that C( ) G L
(
H from
), we an put
H # G Ô p (R ) :
(3.104)
G  G # G, we have ensured that H is Ginvariant. In view of Proposition 3.77(ii), H from (3.104) is the smallest G invariant linear spa e ontaining H from (3.102). Again we an endow H from (3.104) by the norm h H dened by (3.98), whi h indu es a p m ner topology than the relativized topology from Car% ( ; R ) and guarantees (3.93). In parti ular, we an take G # L ( ). Often it is advantageous to have G separable, so that it has a good sense to onsider G smaller than L ( ), f. (3.161) below. p m The orresponding onvex ompa ti ation Y H % ( ; R ) is stri tly ner than the
As
;
onvex ompa ti ation obtained by the hoi e (3.102). To avoid te hni alities, let us show it only for a spe ial ase ality, we an suppose that
# (0; 1)
Without loss of gener
ontains a fun tion g dis ontinuous at some x ò , limxÿx0 g (x) and limx÷x0 g (x) do exist. Let us now take
G
and for simpli ity the limits
m # 1.
and 0
0
0
0
¡ 1, the norm onsisten y follows dire tly from Theorem 3.39(iii). For p # 1 we an rst B % L ( ; Rm ) by the Nemytski mapping N : u ÜÙ uu"" %" (L strong,L %" strong)homeomorphi ally onto the ball B 1 1%" L %" ( ; Rm ) and then to make the onvex om% % " m m q pa ti ation DM r %1 1%" ( ; R ) of this transformed ball, where R # { v ò C (R ); ; v ò R : v ( s ) # R v(ss"" %" )} is a omplete subring of C (Rm ). If " ¡ 0, this latter onvex ompa ti ation is L %" %" m p p # C ( ) norm onsistent by Theorem 3.39(iii) and therefore (DM p N) with H r %1 1%" ( ; R ) ; n H ; i H R % " m q Ô (R ) forms an L norm onsistent onvex ompa ti ation of the ball B % L ( ; R ) whi h m is equivalent with the original onvex ompa ti ation DM R % ( ; R ) via the adjoint mapping to p. N:HÙH 64
If
p
1
transform the ball
/(1
/(
)
1
1
0
;
/(1
1
)
1
/(
)
0
)
1
1
*
;
1
/(
)
1
1
1
;
3.3 A lass of onvex ompa ti ations of balls in
two sequen es {
u k (x) # (k%) 2
1/
p
u1k }kòN
(
Lp
u2k }kòN in B % dened by u1k (x) # (k%)1 p x0 x0%1 k and , where M denotes the hara teristi fun tion of M . Ap/
and {
x 0 "1/ k ; x 0 )
Ë 181
spa es
(
;
/
)
plying the hoi e (3.102), these sequen es annot be separated; more pre isely, they have the ommon limit, whi h is the DiPernaMajda measure from Figure 3.7 (with
a # %1
/
b # 1). On the other h # g0 sp one has
p and
Indeed, for
hand, they an be separated by
x 0 %1/ k X k%g0 (x) dx k Ù x 0
lim i H (u k ); h # lim 1
k Ù while
x0 k%g0 (x) dx X k Ù x 0 "1/ k
lim i H (u k ); h # lim 2
k Ù
H
from (3.104).
# % lim g (x) xÿx0
0
# % lim g (x) : x÷x0
0
These limits has been supposed to be dierent from ea h other. Contrary to Example 3.47, we will not interpret su h stri tly ner onvex ompa ti ation in terms of DiPernaMajdalike measures, though it might be possible.
Example 3.49 (Fonse a's extension of L ( ; Rm )). 1
For
p # 1 one an take
H # h : , Rm Ù R; ;h0 ò C0 ( , S m"1 ) : h(x ; s) # h0 x ; Let us note that
s s Ǳ : s
(3.105)
H Car1 ( ; Rm ) is C( )invariant. A natural norm on H is now
h H #
(
max
x ; s )ò , S m"1
h(x ; s)
(3.106)
whi h generates a ner topology than the relativized topology from
Car% ( ; Rm ) be1
h ¢ % h H . Equipped with this norm, H is isometri ally isoC0 ( ; Sm"1 ) and thus it is separable. 1 m m m Then Y H % ( ; R ) is equivalent with F % ( ; R ) := { òF( ; R ); P , S m"1 (d x d s ) ¢ m %} with F( ; R ) being the set of Fonse a's measures dened by (3.76); the ane homem "1 ) is just the adjoint operator to h ÜÙ h : C ( ; S m "1 ) Ù omorphism H Ù r a( S 0 0 H with h dened by h(x ; s) # h0 (x ; ss"1 )s. 1 m This onvex ompa ti ation is not L norm onsistent.65 Also, F % ( ; R ) is stri tly oarser onvex ompa ti ation of B % than the DiPernaMajda measures 1 m DMR % ( ; R ) provided R is greater than the ring from (3.45).66
ause of the estimate % morphi with ;
*
;
65
To see it, the reader an onsult Remark 3.36 where various sequen es, distant from ea h other in
R
L1 norm, onverge to the same limit when embedded into F( ; m ). m"1 ) is dense in C ( , S m"1 ), the spa e H from (3.105) ontains densely H # 66 As C 0 ( ) C ( S 0 0 m C0 ( ) {v ò C( ); ;v0 ò C(S m"1 ) : v(s) # v0 (s/s)s} whi h is obviously ontains in C( ) m ) Ê Y 1 ( ; m ) Ê Y 1 m) ² 1 Ô (R ) with R from (3.45). Using Theorem 3.40(i), we get F% ( ; H; % H0 ; % ( ;
R
Y1
C( )Ô1 (R); %
(
;
Rm
R
)
Ê DMR % ( ; Rm ). In view of Remark 3.36, even 1
;
R
R F% ; Rm ° DMR % ; Rm (
)
1
;
(
).
182
Ë
3 Young Measures and Their Generalizations
Example 3.50 (Coarser onvex ompa ti ations I).
Let us take
p ò [1; %℄, S #
Rm ,
and
H # L p ( ) (Rm ) ;
Rm
where (
*
)
*
(3.107)
denotes naturally the spa e of linear fun tions
Rm Ù R; note that p
is
p, f. (1.35). Su h H is obviously L ( )invariant. Let us p m note that a tually H Car ( ; R ). Indeed, as every h ò H takes the form h ( x ; s ) # m g id # l#1 g l (x)s l with s # (s1 ; :::; s m ) ò Rm and id : Rm Ù Rm denoting the
the onjugate exponent to
identity on
Rm , we an always estimate by the Hölder inequality:
h% #
m !! ! !!X H g ( x ) u ( x ) d x !!! l l !! !! ! u L p ;R m ¢ % !
l #1
sup (
¢
)
sup
u L p ;Rm (
)
¢%
u L p
(
;Rm ) g L p ( ;Rm )
It is then natural to dene the norm on
# % g L p
(
;Rm )
:
H as
h H # g id H # g L p
(
(3.108)
;Rm )
h # g id ò H . It makes (3.93) satised. Then Y H % ( ; Rm ) is equivalent for 1 p ¢ % with a ball of the radius % ò R% in L p ( ; Rm ) endowed with the weak* m topology67 while for p # 1 it is equivalent with the ball of the radius % in L ( ; R ) Ê m vba( ; R ) endowed by the weak* topology, i.e. the ball in the bidual spa e of L1 ( ; Rm ). The ane homeomorphism is via the mapping H Ù L p ( ; Rm ) adp m joint to the operator g ÜÙ g id : L ( ; R ) Ù H . As the resulting onvex ompa ti ation is equivalent with B % endowed with the weak* topology provided p ¡ 1, we obtained obviously a minimal Hausdor onvex
ompa ti ation of B % . On the other hand, for p # 1 it is not true. To see this, we an adopt another hoi e of H , for example p
for
;
*
*
*
H # C( ) (Rm ) : *
(3.109)
H , we an take again the norm (3.108). For 1 p ¢ %, this hoi e H from (3.109) is dense 1 m in the norm (3.108). However, Y ( ; R ) with H from (3.109) is H%
As for the norm on
does not hange M(F H ; % ) in omparison with (3.107) be ause in
H
from (3.107)
;
% in C( ; Rm ) Ê r a( ; Rm ), whi h is a stri tly
oarser Hausdor onvex ompa ti ation of B % . To see this, we an paraphrase the
onstru tion from Example 3.48: supposing # (0 ; 1) and m # 1 to avoid te hni ali1 2 1 ties, we take two sequen es { u } k òN and { u } k òN in B % dened by u ( x ) # k% x 0 x 0 %1 k k k k 2 and u ( x ) # k% x 0 "1 k x 0 . Applying the hoi e (3.109), these sequen es annot be k equivalent with the ball of the radius
*
(
(
/
;
;
/
)
)
separated; more pre isely, they have the ommon limit, whi h is the Dira measure
67
For
1 p % it is merely the weak topology be ause of the reexivity of L p ( ; Rm ).
3.3 A lass of onvex ompa ti ations of balls in
Lp
spa es
Ë 183
Æ x0 ò r a( ). On the other hand, they an be separated by H from (3.107): indeed, limkÙ # 0 while limkÙ # % ¡ 0 provided h # 0 x0 id.
(
;
)
It demonstrated that the hoi e (3.107) still does not provide a minimal Hausdor onvex ompa ti ation if
p # 1.
It is noteworthy that there exists still stri tly oarser onvex ompa ti ations of the ball in
L1 ( ; Rm ) than the ball in r a( ; Rm ). One of them an be reated by
H # C0 ( ) (Rm )
*
with
C0 ( )
(3.110)
denoting the spa e of ontinuous fun tions vanishing on the boundary
bd( ). This gives Y H % ( ; Rm ) anely homeomorphi with the ball of the radius % in C ( ; Rm ) Ê r a( ; Rm ). 1
;
0
*
Example 3.51 (Coarser onvex ompa ti ations II). pa ti ations an be obtained by mapping
B%
Another lass of onvex om
homeomorphi ally onto a ball (possi
bly of a dierent radius) in another Lebesgue spa e and then ompa tify this latter ball. E.g., if we map
uu""/(1%")
B % L1 ( ; Rm )
onto
B %1
1%")
/(
L
%" ( ;
1
Rm
) via the mapping
uÙ Ü and then ompa tify this latter ball by means of H from (3.107) with p # 1 % ", we obtain the same ee t as if we ompa tify B % by means of H # L1%1
/
"
(
; Rm ) {v}
(3.111)
v ò C1 1%" (Rm ; Rm ) dened by v(s) # ss"" 1%" . It is left as an exer ise to show 1 1 m m that H Car ( ; R ) and, for " ¡ 0, su h hoi e of H yield Y H % ( ; R ) equiva1 1% " 1% " m lent to the ball of the radius % in L ( ; R ) endowed with the weak topol1% " m ogy; the ane homeomorphism H Ù L ( ; R ) is the adjoint operator to g ÜÙ h g "" 1%" . This onvex ompa ti ation is stri tly oarser than with h g ( x ; s ) # g ( x )  s s 1 m Y ( ; R ). with
/(
)
/(
)
;
/(
)
*
/(
Remark 3.52.
Let us note that
)
H from (3.107) is L
(
)invariant while H from (3.109) p ¡ 1. Similarly it
is not, though they generate equivalent onvex ompa ti ations if
H from (3.101) and (3.97), respe tively. It shows that sometimes the fa t H is not Ginvariant for a given C( ) G L ( ) may be only arti ial and an be removed by a suitable enlargement of H , whi h is possible up to the losure of H p in Car% ( ; S) without hanging the reated onvex ompa ti ation; f. Theorem 3.40. On the other hand, sometimes the la k of G invariantness for G greater than C ( ) may be essential, e.g. for the ases (3.109) with p # 1 or (3.102). holds also for
that
Y% ( ; Rm ) is not a norm onsistent ompa ti ap m m tion of the ball of L ( ; R ). Nevertheless, Y % ( ; R ) is T  onsistent if T is the relap m tivized strong topology of L ( ; R ) with p ò [1 ; %). In parti ular, having a bounded
Remark 3.53.
We mentioned that
p
184
Ë
3 Young Measures and Their Generalizations
L
u ÙÆu
; Rm
; Rm ) and also Ù 0 for any p %.
sequen e { k } k òN ( ) onverging weakly* to some PSfrag repla ements ( k) ( ) weakly*, we an laim68 that k L p ( ;Rm )
Æu
PSfrag repla ements
Summary 3.54.
For
1 p
u "u
uòL
(
the relations among the above examples an be dis
played by the diagram (3.107) (3.109)
DiPerna,Majda (refined) (3.104)
DiPerna,Majda
Young
(3.102)
Lebesgue (3.97)
(3.99)
Ê (3.109)
p # 1 this diagram is enhan ed by another row:
(3.45)
while for
DiPerna,Majda (refined) (3.104)
DiPerna,Majda
Young
(3.102)
Lebesgue (3.97)
G # L ( ) and if R ontains (3.45) R ontains (3.45) finitely additive Radon measures Fonseca measures (3.107) on
(3.109)
L1%"
(3.111)
if
if
Radon measures on (3.110)
(3.105)
where ea h arrow goes from a ner onvex ompa ti ation to a oarser one. Moreover, ea h terminal onvex ompa ti ation is stri tly oarser than the initial one ex ept the
ase (3.104)
Ù(3.102) if G # C( ). Besides, no other arrow an be added; it means the
onvex ompa ti ations, whi h are not onne ted by a hain of arrows, are a tually
p # 1 and R given by (3.45), the relation between the 1 DiPernaMajda, Fonse a, and L Young measures is pretty exa t in the sense that69 not omparable. Moreover, for
DMR % ( ; Rm ) # sup F% ( ; Rm ); Y% ( ; Rm )
1
1
(3.112)
;
with F(
; Rm )
being dened by (3.76) but with
repla ed by
. In
fa t, we proved
only some of these relations, the rest being left as an exer ise.
Remark 3.55 (Convex ompa ti ations of L balls are universal). Let us note that "p p S ' : h ÜÙ h , dened70 by h (x ; s) # h(x ; s s S ), is an isometri al isomorphism p Car% ( ; S) Ù Car% p ( ; S); indeed, one has the identity 1
(1
)/
1
h% #
!! ! !!X h ( x ; y ( x )) d x !!! !! !! ! y L p ;S ¢ % !
sup (
)
where the initial seminorm on erns
Car ( ; S). 1
68
#
!! ! !!X h ( x ; y ( x )) d x !!! ! !! ! p ! y L 1 ;S ¢ % !
sup (
# h% p ;
)
Carp ( ; S)
Then, having some linear subspa e
while the terminal one on erns
H Carp ( ; S), H # S ' (H)
This is, in fa t, a well known result; see, e.g., Da orogna [241, Corollary 6.2℄ or Málek et al. [512,
Thm. 2.91℄.
69
p
The formula (3.112) follows from Theorems 3.40(i) and 3.42 if one realizes that, for p # 1, C ( ) R ) is dense in H % C ( ) C p ( m ) with H as in (3.105) but with C ( , S m"1 ) in pla e of C 0 ( , S m"1 ). (1" p )/ p Cf. also (3.185) with S1 # S2 # S and ' ( x ; s ) # s s . S
Ô (
70
R
3.3 A lass of onvex ompa ti ations of balls in
Lp
spa es
Ë 185
Car ( ; S) determines a onvex ompa ti ation Y pH % p ( ; S) of a ball B % p L ( ; S). The adjoint mapping ( S ' H ) : H Ù H is a homeomorphism and maps the onvex p p ( ; S); this means
ompa ti ation Y H % ( ; S) of the ball B % L ( ; S) onto Y p H %p 1
1
1
;
*
*
*
1
;
;
p Y H ; % ( ; S) ;
H ; iH
Ê
H ; ip H N'
Y p p ( ; S) ; H;% 1
*
*
H # S ' (H) and N' : L p ( ; S) Ù L1 ( ; S) is the Nemytski mapping generated 1" p p . by ' dened by ' ( x ; s ) :# s s S
where
(
)/
Thus there is a onetoone orderpreserving orresponden e between onvex om
p
B % L p ( ; S) of the form Y H % ( ; S) with H ò Carp ( ; S) and 1
onvex ompa ti ations of a respe tive ball, namely B % p L ( ; S), of the form 1 1 Y H % p ( ; S) with H ò Car ( ; S). Thus, the rst diagram in Summary 3.54 an be embedded into the se ond one provided " # p " 1, its image being denoted by gray pa ti ations of a ball
;
;
boxes.
Remark 3.56 (Convex ompa ti ations of Orli z spa es). Let us onsider an in reas% % ing onvex ontinuous fun tion M : R Ù R su h that M (0) # 0, lim a Ù% M(a) # %, and, for some k ; a ¡ 0 and every a £ a , M(2a) ¢ kM(a). The subset
0
0
L M ( ) # of
L1 ( )
uòL
1
(
);
X M ( u ( x )) d x
%
is alled an Orli z spa e.71 If equipped with the so alled Luxemburg norm
M(u(x)/r) dx ¢ 1}, it be omes a separable Bana h spa e. The u ò L M ( ); P M(u(x)) dx ¢ 1}. In parti ular, for p M M(a) # a we have obviously L ( ) # L p ( ). The purpose of this generalization is to handle nonlinearities with nonpolynomial growth, as M ( a ) # (1 % a )log (1 % a ) " a or M(a) # a p (1%log(a)). In parallel with the theory for L p spa es, we an dene here
u L M
(
)
unit ball
# inf {r ¡ 0 : B1
P
is then just equal72 to {
the relevant spa e of integrands
CarM ( ; Rm ) # h : , Rm Ù R Carathéodory; ; a h ò L ( ); b h ò R% : !!!!h(x ; s)!!!! ¢ a h (x) % b h M s 1
!
!
h # sup u LM ¢% !!!!P h(x ; u(x)) dx!!!! ; note that h u is a 1 1 tually integrable be ause h u ¢ a h % b h M ( u ) and both a h ò L ( ) and M ( u ) ò L ( ) M M provided u ò L ( ). For a subspa e H Car ( ; R), we an dene a onvex ompa tiM M ation Y H 1 ( ; R) ; H ; i H
, where the embedding73 i H : L ( ) Ù H is dened equipped with the seminorm %
(
)
*
*
;
71
Su h spa es were introdu ed in thirties by Orli z [583℄. More details an be found, e.g., in mono
graphs by Appell and Zabrejko [23℄, Krasnoselski and Ruti ki [441℄, and Kufner, John and Fu£ík [467℄.
72
See Krasnoselski and Ruti ki [441, Thm. 9.5℄.
73
By the ontinuity of the Nemytski mapping
Thm. 17.6℄, the embedding
iH
Nh : L M ( )
is (strong,weak*) ontinuous.
Ù L ( ) for any h ò Car M ( ; R), f. [441, 1
Ë
186
3 Young Measures and Their Generalizations
i u ; h> # P h(x ; u(x)) dx and Y HM 1( ; R) is the weak* losure of the unit M ball B 1 in L ( ) embedded via i H . Likewise in Remark 3.55, we an dene the isometriM 1 "1 (s)/s).
al isomorphism Car ( ; R) Ù Car ( ; R) by h ÜÙ h with h ( x ; s ) # h ( x ; sM M Then S is a homeomorphism between the onvex ompa ti ation Y H 1 ( ; R) of the 1 M unit ball B 1 in L ( ; R) and the onvex ompa ti ation Y p ( ; R) of the ball again by < H ( )
;
*
;
BM 1 (
H ; M (1)
)
in
L1 ( ). In parti ular, it shows that Y HM 1 ( ; R) is onvex and ompa t in H
*
;
.
Example 3.57 (Extension of a norm). Assuming 1  p ò H Carp ( ; Rm ), we an p m extend the norm on L ( ; R ) ontinuously by
p y < ; 1
:#
p
 >
R
m );
¢ %}
(3.113)
() #
This ts with the abstra t Example 2.29 with
p Y H ( ;
:
p are onvex and equal to Y H ; % ( ;
R
p
() . In parti ular, {
ò
m ). This is in parti ular the
ase of (3.28) and (3.30).
3.3.d
Coarse polynomial ompa ti ation by algebrai moments
H # G V with a niteV omposed from polynomials of the order ¢ 2k. The generalized p m Young fun tionals ò Y ( ; R ) then orresponds to their algebrai moments, i.e. H Sophisti ated onstru tions exist for the spe ial ase of dimensional spa e
m # Ǳ (1 s 1 s 2    s mm ) 1
(3.114)
2
# (1 ; 2 ;    ; m ) is the multiindex of nonnegative integers 1 % 2 %    % m ¢ 2k. Namely, for any h ò H , it holds that
where
#
; h
H ; g ¢2 k
s 1 s 2    s mm # 1
2
H X g (x) ¢2k
:#
su h that
m (x) dx :
(3.115)
This potentially gives a han e to work e iently with su h oarse onvex
 ompa ti
ations. Yet, to this goal, one needs an e ient hara terisations of these moments.
m # 1. m £ 1, denoting m # (m ) ¢2k , we dene the so alled Henkel
This is not trivial and satisfa tory haraterisation exists only for In the general ase matrix
Hk m (
) as
Hk m :# m1%1 (
)
; ;
m % m
0 0
¢ 1 %    % m ¢ k : ¢ 1 %    % m ¢ k
(3.116)
It is used parti ularly e iently in the onedimensional situations where
m% ℄k #
plies to [
;
1
Hk m (
) sim
:
Lemma 3.58 (Polynomial moments of probablity measures).74 It holds
l m # X s i d (s) R
74
i #0 ; 1 ; : : : ; 2 k
;
òr a% (R) # m ò R k ; 2
1
Hk m £ 0 : (
)
For this lassi al result see e.g. the monograph J.A. Shoat and J.D. Tamarkin [727℄.
(3.117)
3.3 A lass of onvex ompa ti ations of balls in
Lp
spa es
Ë 187
H£0
The ordering of matri es in (3.117) is the so alled Löwner ordering, i.e. {
}
is the losed one of all positive semidenite matri es. Disregarding the restri tion on balls in this subse tion, Lemma 3.58 gives:
Proposition 3.59 (Polynomial onvex  ompa ti ation of L p ( )). Let V C k (R) i be the linear hull of { s ; i # 0 ; 1 ; :::; 2 k } and H # C ( ) V . Then, for p ¡ 2 k , the p p
onvex  ompa ti ation Y H ( ; R) of L ( ) is equivalent to 2
m # (mi )i#
0;1; : : : ;2
k;
mi ò L p i ( ); Hk (m(x)) £ 0 : x ò : /
(3.118)
a.a.
To relax oer ive optimization problems, the hara terisation (3.118) is needed for
p # 2k, in whi h ase it holds for pnon on entrating 's. Also the following onsequen e is useful:
Corollary 3.60. Let '(t) # #k t be a one dimensional,
k ¡ 0. Then, any solution of the semidenite program: 2
0
oer ive polynomial, i.e.
2
Minimize
2k H #0
m#
m
Hk m ³ 0 with m # 1 and m # a ;
subje t to
(
)
0
is omposed of the algebrai moments of a measure
(3.119)
1
solving the following abstra t
optimization problem dened in measures: Minimize X
R
Conversely, if
'(s)
ds)
(
subje t to X
R
ds) # a ;
s
(
òr a% (R) :
(3.120)
1
solves (3.120), then its algebrai moments solve (3.119).
Remark 3.61 (Ve torial problems).
The ve torial situation
m ¡ 1
is unfortunately
mu h more ompli ated and an be handled only approximately, using the asymptoti s for ountable number of higherorder momenta. To this goal, as devised for global optimization of polynomials with moments [380, 476, 477, 480, 594℄, one is to use also the so alled lo alizing matrix
Lk m # m1%1 (
2
)
; ;
m % m "
m1 %1%
2 ; ;
Lk m (
) dened, for some
m % m "   
" m 1 % 1
; ;
¡ 0, as
m % m %2
0 0
¢ 1 %    % m ¢ k"1 : ¢ 1 %    % m ¢ k"1
u ò L ( ; Rm ); u(x) ¢ % :a.a. x ò } H # C( ) V with V # span{s11 s22    s mm ; # (1 ; 2 ;    m ) ¢
Then, the onvex ompa ti ation of the ball {
Y H ( ; R 2k ; s ¢ %} an be approximated for û Ù by the onvex sets in
m ) with
M # (m ) ¢2k ; ;(m )2k%1¢ ¢2 : û
:a.a. x ò : m
(0 ; 0 ; : : : ; 0)(
û
m ò L ( ; Rm )
x) # 1;
H m x £ 0; û(
( ))
with
onsidered for û simple
M
û
for
# 0; :::; 2û;
and
L m x £0 û(
m # (m ) ¢
2û
( ))
(3.121)
£ k. In view of Proposition 3.59, the situation for m # 1 is parti ularly M # M k for any û £ k. The usage
is independent of û and, in parti ular,
û
Ë
188
3 Young Measures and Their Generalizations
of su h result is for optimal ontrol problems with apriori bounded admissible ontrols or for oer ive variational problems after an approximation by a dis retisation of
whi h, for a xed dis retisation, an be expe ted to have solutions in su iently
big
L
balls. Then û is to be taken su iently big and represents another approxima
tion parameter, realizing an outer approximation of the semidis retised problem. Su h approximated semidis retised problems lead to a semidenite mathemati al
programming (SDP) for whi h e ient numeri al methods and software pa kages exist, f. [305, 380, 381, 429, 799℄.
3.3.e
Compatible systems of Young fun tionals on B(I; L p )
The above onstru tions are appli able rather to stati problems or evolution problems whi h are in some sense quasistati with spe ial properties, f. Remark 8.7. For general evolution problems, an interesting and elegant onstru tion takes into a
ount ertain nonlo al intera tions like we already presented on an abstra t level in Se t. 2.6, although it should openly be said that its appli ability and interpretation is rather doubtful be ause of too big generality. Nevertheless, it develops a su iently wide lass of onvex ompa ti ations of the spa e of bounded mappings
I Ù L p ( ; Rm ) with I a ompa t interval of R, denoted by B(I; L p ( ; Rm )). Having in
mind some uniform apriori estimates usually available, it su es to ompa tify only the ball
B :# u òB(I; L p ( ; Rm )); : t ò I : u(t) L p
(
;Rm )
¢ :
(3.122)
Coarse ompa ti ations handle spatial os illations/ on entrations on parti ular time levels separately but ner ompa ti ations an handle possible orrelations of su h os illations/ on entrations at various time instan es. Always, a nite (although not apriori given) number of those time instan es su es to be in orrelation. To this goal, we systemati ally exploit the theory of inverse systems of onvex ompa ti a
π # (t1 ; t2 ; :::; t# π ) a nite partition of I R t1 t2 ::: t# π where #(π) denotes the number of elements of π and where t i ò I for all i # 1; :::; #(π). Let us denote by F(I) the olle tion of all su h partitions ordered by in lusion. It makes F( I ) dire ted. Let us further take, for any π ò F( I ), some p # π ,m ). Let us further dene normed linear subspa e H π Car ( ; R
tions from Se tion 2.6. Let us denote by with
(
(
)
)
(
eπ : B(I; L p ( ; Rm )) Ù Hπ *
eπ (u); h
)
by
:# X h(x ; u(t ; x); u(t ; x); :::; u(t# π ; x)) dx :
1
2
We onsider simply the Cartesian produ t AπòF( I )
Hπ *
Hπ by (eπ )πòF I *
( )
Hπ . Then we embed B *
and dene
YH; (I; L p ( ; Rm )) :# l eπ (B )πòF(I) ; p
(3.123)
equipped with the Tikhonov
produ t topology here ounting the weak* topologies of ea h into AπòF( I )
)
(
(3.124)
3.3 A lass of onvex ompa ti ations of balls in
Lp
spa es
Ë 189
where the losure refers to the Tikhonov produ t topology ounting the weak* topolo
Hπ and where H abbreviates the olle tion (Hπ )πòF(I) . Also, p p m )) # the losure of e ( B ), whi h is a ompa t we an onsider Y (I; L ( ; π Hπ ; p * #(π),m ) dened by (3.86). Dening the subset of H π . It is exa tly the set Y ( ; Hπ ; p m )) Ù L p ( ; #(π), m ) Ê L p ( ; m )#(π) by j ( u ) # mapping j π : B( I ; L ( ; π *
gies on ea h parti ular
R
R R
R
R
u(t1 ; ); :::; u(t# π ; )), we an see that eπ # i Hπ jπ with i Hπ dened in (3.84) with p Hπ in pla e of H . As the triple (Y Hπ ( ; R# π ,m ); i Hπ ; Hπ ) forms a onvex ompa tm p p p # π ,m ); max i ation of the set { u ò L ( ; R i #1 # π P j#1 u ij (x) dx ¢ % } just as (
(
)
)
(
*
;
)
(
p p explained in Se tion 3.3, ( Y H ; ( I ; L ( ; π
Rm
;:::;
)
(
; eπ ; Hπ ) makes a onvex ompa ti ation of B . If H π are ri h enough (as, e.g., in Theorem 3.39(iii)) ea h i H π is inje tive but e π is not be ause j π is not inje tive (ex ept a trivial ase that I itself is nite). Let p p m us note that, in spite of it, ( e π )πòF I is inje tive. Also note that Y H ( I ; L ( ; R )) p p AπòF I Y H ( ; R# π , m ) and, as ea h Y H ( ; R# π , m ) is ompa t, by Tikhonov's π π p # π ,m ) and thus also Yp (I; L p ( ; Rm )) itself is omTheorem 1.5, AπòF I Y Hπ ( ; R H p p m pa t, too. Hen e, (Y ( I ; L ( ; R )) ; i ) forms a ompa ti ation of B . This ompa tH
*
))
( )
(
( )
;
(
)
;
)
;
(
( )
)
;
;
;
i ation is not metrizable ex ept trivial ases.
π1 Let us now assume existen e of the olle tions ( π2 )π1 ; π2 òF( I ) of linear operators
P
satisfying (2.44) with
Hπ in pla e of Fπ .
1 :# [Pπ π2 ℄ : H π1 Ù H π2
π1 π2
*
: Y Hp π
π1 π2
is surje tive, and then also
1 ;
surje tion, just showing that
p
Y Hπ
1 ;
(
By (2.44a), the adjoint mapping
(
*
)
(3.125)
p
; R# π1 ,m ) Ù Y Hπ (
p
; R# π1 ,m ); Hπ1 ; eπ1 ³ Y Hπ (
*
*
2 ;
(
)
2 ;
(
; R# π2 ,m ) (
)
; R# π2 ,m ); Hπ2 ; eπ2 ; (
)
*
is a
(3.126)
R
p
#(π1 ),m ) of B is ner than the onvex om( ; % 1 ; p π1 # (π2 ), m pa ti ation Y H ). By (2.44a), is
ontinuous, and by (2.44b) it sat( ; π2 π2 ; π2 π1 π1 π ises π3 π2 # π3 , and eventually (2.44 ) ensures π # identity. The olle tion p p m )) then satises the property that ( π )πòF( I ) of Y (I; L ( ; H; i.e. the onvex ompa ti ation
R
Y Hπ
R
π2 #
π1 π2 π1 whenever
f. (2.47). Altogether, the operators
p
S # (Y Hπ
1 ;
(
π1 π2 :
π1 π 2 play the role of the bonding mappings and thus
; R# π1 ,m ); Hπ1 ; eπ1 ); (
)
*
π1 π2 π1 ; π2 òF( I ); π1 π2
is an inverse system (in the sense of Se t. 1.1) of onvex ompa ti ations of
YH; (I; L p ( ; Rm )) is its limit, i.e. p
YH; (I; L p ( ; Rm )) # lim S p
By Proposition 2.39,
e
ØÚÚ
YH; (I; L p ( ; Rm )) p
with
S
(3.127)
B , and
from (3.127).
itself is a onvex ompa ti ation of
B ; the
embedding is ( π )πòF( I ) and the linear spa e indu ing its onvex stru ture is now
Ë
190
AπòF( I )
3 Young Measures and Their Generalizations
Hπ . By (2.39), also *
YH; (I; L p ( ; Rm )) Ê sup Y Hπ ; ( ; R#(π),m ): p
p
πòF( I )
p p H; ( I ; L ( ;
The threads, i.e. the elements of Y
Rm
)), f. (1.3), are also alled
systems of Young fun tionals. For a spe ial hoi e of the system
ompatible
H as in Example 3.63
p # 1, su h systems have been invented in [248, Se t.7℄ (under the name
below and
ompatible systems of generalized Young measures) and further used in [249, 250, 306,308℄. A general ansatz based on Example 3.63 below has been s rutinized in some variant also in [466℄.
Example 3.62 (Non orrelated threads).
A rather standard but oarse onvex om
pa ti ation is obtained simply by opying the onstru tion from Se tion 3.3 onstantly at ea h time instant, obtaining thus the onvex ompa t subset of the produ t (
H
*
I ) into whi h
B%
is embedded simply by
p
YH
;
% ( ;
Rm
)
I
u ÜÙ (i H (u(t)))tòI . Up to an
equivalen e of onvex ompa ti ations, we an obtain this onvex ompa ti ation in the above framework, too. To this goal, let us put
Ü H#k#π1 h k (x ; s k ); h k ò H Ǳ : Hπ :# (x ; s1 ; :::; s# π ) Ù (
(
)
(3.128)
)
Carp ( ; R# π ,m ) if H is a subspa e of Carp ( ; Rm ). # π and we an dene π Thus, due to the spe ial hoi e (3.128), here H π Ê ( H ) Ê ( H ) I the linear inje tive mapping : (H ) Ù AπòF I Hπ by ÜÙ (π )πòF I whi h is also a homeomorphi al embedding. Dening the system H :# ( H π ) πòF I by taking p Hπ from (3.128), we obtain the onvex ompa ti ation YH (I; L p ( ; Rm )) by (3.124). π1 π1 The bonding mappings are dened as π2 with P π2 : H π2 Ù H π1 given by Hπ
Obviously,
(
is a subspa e of
)
*
*
*
*
(
)
*
( )
( )
( )
;
π Pπ12 h(x ; s1 ; :::; s# π1 ) :# h(x ; s j1 ; :::; s j# π2 (
)
(
)
(3.129)
)
j : π2 Ù π1 is just the in lusion π2 π1 . The mapping rep p Y H % ( ; Rm )I then realizes the homeomorphism between Y H % ( ; Rm )I and p YH (I; L p ( ; Rm )), whi h makes these onvex ompa ti ations equivalent to ea h
where here stri ted on
;
;
;
other.
Example 3.63 (Correlated threads based on DiPernaMajda's measures).
Based
on
the DiPernaMajda measures, the onstru tion of threads nontrivially orrelated have essentially been invented in some variant in [466℄. For
p DMR #π (
)
(
; R
#(π),m ) with a separable ring R #(π)
C
0
R
π ò F(I),
one an use
(
#(π),m ) orresponding either
the onepoint Alexandro ompa ti ation of the ompa ti ation by a sphere
S# π ,m"1 . The bonding mappings are again determined as in Example 3.62 by means of (3.129). Here, for any π2 π1 ò F( I ), it is important that (
)
:g ò C( ) :v ò R # π2 : (
with Ô
)
p p # π1 1 Pπ π2 ( g Ô ( v )) ò C ( ) Ô (R
p from (3.89), whi h indeed holds true for (3.129).
(
)
)
(3.130)
3.4 A lass of onvex
 ompa ti ations of
Remark 3.64 (Threads with a bounded variation). d(s1 ; s3 ) ¢ d(s1 ; s2 ) % d(s2 ; s3 )
:π ò F(I); #(π) £ 2 :
dπ
Ë 191
spa es
The onstru tion from Se t. 2.6 an
Rm Ù R% , i.e. the triangle inm is satised for all s ; s ; s ò R . If
be applied here when onsidering the distan e equality
Lp
d :
2
(
)
1
: (x ; s ; :::; s# π ) ÜÙ 1
(
)
2
3
#(π)
H d ( x ; s i "1 ; s i ) ò H π i #2
;
(3.131)
f. (2.49), like in [248, Def.8.1 and 8.6℄ we an dene the dissipation of a thread
I
with respe t to the distan e
d
( )
#(π)
{
;
}
(
Rem.8.3℄, one an write also
dòH
is
; dπ > # i#2 ; note that al
from (3.131). It holds < π
ways
over
Dissd (; I) :# supπòF I where dπ
by
{
)
{
( )
;
}
t i"1 ; t i } due to (3.131). The Helly sele tion prin iple as in Proposition 2.41
an be applied here for sequen es of
's. The ase p ¡ 1 is however not ompatible
with (2.53). Thus a weakened variant of both (2.53) has naturally to be used, based on
pnon on entrating threads. We all a thread ò YH (I; L p ( ; Rm )) pp m non on entrating if there is a net { u } ò B % B( I ; L ( ; R )) attaining the thread p su h that {u (t ; ) ; ò ; t ò I} is relatively weakly ompa t in L1 ( ). Analogously, p p m a sequen e of threads { k } k òN Y H ( I ; L ( ; R )) is alled equi p non on entrating p
the notion of
;
;
if there are nets { {
u k } ò k B % B(I; L p ( ; Rm )) attaining the parti ular k
su h that
u k (t ; )p ; ò k ; t ò I; k ò N} is relatively weakly ompa t in L1 ( ). For a modied
Helly prin iple then see [688, Prop. 6℄.
3.4
A lass of onvex
 ompa ti ations of L p spa es
In this se tion we will join the results from Se tion 3.3 with the theory of onvex
 ompa ti ations
of normed linear spa es as presented in Se tion 2.3 in order
 ompa ti ations of the Lebesgue L p ( ; S) with S a separable Bana h spa e. These onvex  ompa ti ations
to onstru t a su iently ri h lass of onvex spa es
will be sometimes also lo ally ompa t,
B  oer ive, and norm onsistent.
Through
out this se tion, we will onsider
U # L p ( ; S) endowed by the norm bornology
(3.132)
B.
We will onsider again the spa e
Carp ( ; S) from Se tion 3.3, but here we endow
it by the olle tion of seminorms {  % } % òN dened again by
whi h makes
!!
!!
h% # sup !!!!X h(x ; u(x)) dx!!!! ; uòB % !
!
(3.133)
Carp ( ; S) a lo ally onvex spa e. We will refer to this topology as the
natural one. Obviously, it is the oarsest topology whi h makes all the identities
Ë
192
3 Young Measures and Their Generalizations
Carp ( ; S) Ù Carp% ( ; S) with % ò N ontinuous; note that (3.133) oin ides with (3.82). p p The mapping : Car ( ; S) Ù C ( U B ) and the embedding i : U Ù Car ( ; S) are dened respe tively by (natural extension of) (3.83) and (3.84); i.e. < i ( u ) ; h > # [ h ℄( u ) # P h ( x ; u ( x )) d x . Let us note that h % ¢ h % % for any % ò N. Also note that
p h % # h B % # h C 0 B % so that is a homeomorphi al embedding of Car ( ; S) p into C ( U B ) if one onsiders an appropriate fa tor spa e, namely Car ( ; S)/Ker ; *
1
(
re all that
)
C(UB ) was endowed with the olle tion of seminorms  B % %òN .
For a linear subspa e
H Carp ( ; S)
we dene
FH
C ( U B )
again by an
extension of (3.85), i.e.
# (H) % { onstants on U}:
FH
(3.134)
e H : U Ù C(UB ) and i H : U Ù H are dened as in Se tion 3.3, i.e. e H (u) # e(u)FH and i H (u) # i(u)H . Eventually, we put *
Also
p
YH
;
#
% ( ; S)
i
l H * H (
B % ) ;
(3.135)
and
p
YH ( ; S) #
p ℄ YH ; % ( ; S) % òN
p # b lB H i H ( L ( ; S)) :
(3.136)
*
Convention 3.65 (Generalized Young fun tionals).
p
The elements of YH (
; S) will be ad
dressed as generalized Young fun tionals. Let us note that this onvention agrees with the previous Convention 3.38 be ause
p
H endowed with the topology of the seminormed spa e Car% ( ; S) is a subp spa e of the dual of H endowed with the (relativized) topology of Car ( ; S) (or by any p ner lo ally onvex topology), and it is easy to see that Y H % ( ; S) from Se tion 3.3 an p p be a tually onsidered as a subset of Y ( ; S), oin iding obviously with YH % ( ; S) H
the dual of
;
;
dened by (3.135). This justies our notation.
Theorem 3.66.
Let
H be a linear subspa e of Carp ( ; S), p ò [1; %℄, U
B the norm bornology. Then: The linear subspa e F H of C ( U B ) is B  onvexifying ( f. * (M(F H B ) ; F ; e H ) is a onvex  ompa ti ation of ( U; B ). H
and
FH
given
by (3.132) and (3.134), and (i)
(2.16)) and thus
Carp ( ; S) su h that H H , then M(FH B ) ³ M(FH B ), and if H has the same losure in Carp ( ; S) as H , then M(FH B ) lo Ê
(ii) If
H
is another linear subspa e of
FH B ).
M(
H is endowed with a lo ally onvex topology ner than the natural topolCarp ( ; S), then (YHp ( ; S); H ; i H ) forms a onvex  ompa ti ation of ( U; B ) whi h is equivalent with (M(F H B ) ; F ; e H ) via the adjoint mapH ping .
(iii) Moreover, if
*
ogy indu ed from
*
*
Proof. By Theorem 3.39, every Therefore,
FH
is
FH B %
#
FH ; %
is a onvexifying subspa e of
C0 (B % ).
B  onvexifying with respe t to the anoni al norm bornology base.
3.4 A lass of onvex
By (3.134), (
FH
 ompa ti ations of
ontains onstants, so that M(F H B ) is a onvex
U; B ) by Theorem 2.22, whi h proves (i). As for the point (ii), obviously F H
FH
provided
HH
Lp
spa es
Ë 193
 ompa ti ation of
, and therefore M(F H B )
² M(FH B ) again by Theorem 2.22. p As the natural topology of Car ( ; S) is proje tively indu ed from C B ( U ) via , the fa t that lCarp S H # lCarp S H implies l C B U ( H ) # l C B U ( H ), whi h implies lo
l C B U F H # l C B U F H , whi h eventually implies M(F H B ) Ê M(F B ) by TheoH
;
(
(
)
)
(
(
;
(
)
)
(
)
)
rem 2.22. Let us go on to the point (iii). As the topology on
H
is ner than the topology in
Carp ( ; S) proje tively via from C(UB ), the linear operator : H Ù FH is ontinuous. The fa t that the adjoint operator : FH Ù H restri ted on M # { ò FH ; # 1} is inje tive and has a weakly* ontinuous inverse an be demonstrated exa tly as in the proof of Theorem 3.39. As M(F H B ) M by the very denition of M(F H B ) and e H # i H by (3.87), we an on lude that realizes p Å the ane homeomorphism between M(F H B ) and Y ( ; S). H
du ed on
*
*
*
*
*
*
Proposition 3.67 (Lo al ompa tness, onsisten y). Let H be a linear subspa e of Carp ( ; S) endowed with a lo ally onvex topology ner than the topology indu ed from Carp ( ; S). Then: m (i) If S # R and H L p ( ) (Rm ) ; (3.137)
then (ii) If
*
YH ( ; Rm ) is sequentially B  oer ive. p
p % and H ontains a oer ive integrand h in the sense H ó h ;
where
p
h (x ; s) £ s S ;
(3.138)
p
YH ( ; S) is B  oer ive and lo ally ompa t. Moreover, if there is an equality in p m m (3.138) and if also p ¡ 1, S # R , and (3.137) is fullled, then YH ( ; R ) is norm onsistent; i.e. the embedding i H is (strong,weak*)homeomorphi al. then
Proof. Supposing (3.137) and taking a sequen e { u k } k òN su h that { i H ( u k )} k òN on
, we obtain in parti ular that {< i H ( u k ) ; g v >} k òN onverges in m
) and v(s) # m l #1 v l s l for some ( v 1 ; :::; v m ) ò R . As < i H ( u k ) ; g v > # p m P g ( x ) u k ( x ) d x with g ò L ( ; R ) given by [ g ( x )℄ l # g ( x ) v l , we an see that { u k } k òN
p m m
onverges weakly* in L ( ; R ) if p ¡ 1 or in L ( ; R ) Ê vba( ; Rm ) if p # 1. p m In any ase, we have a weak* onvergen e in a dual to the Bana h spa e L ( ; R ) and, by the Bana hSteinhaus prin iple, the sequen e { u k } k òN must be bounded in L p ( ; Rm ); for p # 1 we used also of the oin iden e on L1 ( ; Rm ) of the norms of L ( ; Rm ) and of L1 ( ; Rm ). Thus (i) is proved. Let us take f # h ò F H with h from (3.138). Then
verges weakly* in
R
p for any g ò L
H
*
(
*
*
f (u) #
X h ( x ; u ( x )) d x
£
p X u ( x ) S d x
# u pLp S : (
;
)
Ë
194
3 Young Measures and Their Generalizations
Therefore we have the oer ivity
inf uòU
\
rem 2.22 we an on lude that M(F H B ) is
p
p
£ % p Ù % for % Ù %. By Theo
B % f (u)
B  oer ive and lo ally ompa t, hen e so
YH ( ; S) be ause M(FH B ) Ê YH ( ; S) by Theorem 3.66 and be ause B  oer ivity and lo al ompa tness are invariant under the equivalen e of onvex  ompa ti ations, f. Proposition 2.20. The inverse ontinuity of the embedding i H was shown in
is
Å
Theorem 3.39(iii).
N 2
Corollary 3.68.
Let p ò [1 ; %). There are at least 2 lo ally ompa t B  oer ive on ompa ti ations of L p ( ; S) whi h are even norm onsistent provided additionm m N lo ally ally S # R and p ¡ 1. Besides, if S # R with m ¡ 1, there are at least 2 p m
ompa t B  oer ive onvex  ompa ti ations of L ( ; R ) whi h are even sequenvex
tially lo ally ompa t. Proof. It su es to take
p [Ô v ℄( s ) that
p
YH
s
H # H(s) # C( ) Ôp (R (s)) from (3.88) with Ôp v dened by
# v(s)(1 % s Sp ); f. (3.89). Sin e always 1 ò R (s), we have (3.138) fullled, so p ( ; S) is a lo ally ompa t B  oer ive onvex  ompa ti ations of L ( ; S)
( )
by Proposition 3.67. If
S#
Rm and p ¡ 1, then these onvex  ompa ti ations are
norm onsistent by Proposition 3.67(ii). For dierent
s ò S m"1
 ompa ti ations ( f. again the m ¡ 1 the sphere S m"1 ontains at least 2N ele
we get dierent onvex
proof of Theorem 3.41. However, for
ments. As the above subrings are separable, the respe tive ompa ti ations of
Rm
are metrizable, and the sequential lo al ompa tness follows by the arguments of Ex
Å
ample 3.70 below.
The reader an easily verify that, having a linear subspa e
L
(
G su h that C( ) G
), the property of H to be Ginvariant an be dened again by (3.91) and Proposip YH ( ; S) need not be bounded in
tion 3.43 is still relevant. The only dieren e is that
H
*
be ause, instead of (3.95), we have at our disposal only the estimate
i H (u) H ¢ sup *
h H ¢1
[
h℄(u) ¢ C % sup
h % ¢1
[
h℄(u) ¢ C %
(3.139)
u ò B % , where C % denotes here the onstant from the assumed estimate h % ¢ C % h H . Of ourse, a blowup C % Ù with % Ù is not ex luded, see Exp ample 3.76 where C % # % . Therefore, the joint ontinuity of ( h ; ) ÜÙ h Ǳ stated in p Proposition 3.43 is relevant for ranging only Y H % ( ; S), whi h is ertainly a bounded subset of H . provided
;
*
Let us now have a look how the examples from Se tion 3.3 an be modied.
Example 3.69 (L p Young measures).
Let us take
p ò [1; %), S #
Rm , and H from
(3.97) endowed with the norm (3.98). This strong topology is ner than the (relativized) natural topology indu ed of
Carp ( ; Rm ), whi h an be seen from the estimate (3.99)
valid for any
is
% ò N. Again H
C( )invariant, satisfying also (3.93). By Example 3.46
3.4 A lass of onvex
Lp
 ompa ti ations of
Ë 195
spa es
and Proposition 3.22,
Y H ( ; Rm ) lo Ê Y L1 p
p
(
; C 0 (Rm ))
(
; Rm )
Ê Yp ( ; Rm ) #
ò Y( ; Rm );
X
Rm
s
p
x (d s ) ò L
1
(
) :
(3.140)
p ¡ 1, then also Yp ( ; Rm ) lo Ê Y H1 ( ; Rm ) with H1 # H % L p ( ) m (R ) . By Theorem 3.66 it su es to show that H is dense (in the natural topology of Carp ( ; Rm )) in H1 . Indeed, having some h # g idò L p ( ; Rm ) L(Rm ; Rm ), we an m take always a sequen e h k # g k id with g k ò C ( ; R ) onverging to g in the norm of p m m p L ( ; R ) be ause the embedding C( ; R ) L ( ; Rm ) is dense if p ¡ 1, and then p
Moreover, if
*
by the Hölder inequality
h k " h% #
!! !!X h ( x ; u ( x )) k !! u L p ;R m ¢ % !
sup (
¢
!
)
sup
u L p ;Rm
u L p ;Rm
¢
!!
" h(x ; u(x)) dx!!!!
(
X
)
¢%
)
¢%
sup (
g k (x) " g(x)  u(x)dx
g k " g L p
(
;R m )
u L p
(
Ù 0:
;Rm )
h k Ù h in Carp ( ; Rm ). On the other hand, ea h h k lives in H from (3.97) provided p ¡ 1 be ause we an ertainly write h k ( x ; s ) # g k ( x )  s # m l #1 [ g k ( x )℄ l v l ( s )(1% s p ) with v l ( s ) # s l /(1% s p ), and obviously v l ò C 0 (R m ) provided p ¡ 1. As (3.137) is obviously fullled for H 1 , the onvex  ompa ti ation p Y H1 ( ; Rm ) is sequentially B  oer ive. However, Yp ( ; Rm ) itself is not sequentially This just shows that
B  oer ive.75
Example 3.70 (The generalization by DiPerna and Majda).
Let us take
p ò [1; %), S #
Rm , R a omplete subring of C Rm , and H from (3.102). Again we an endow H by 0
(
h
)
the norm H dened by (3.98), whi h satises (3.93) and indu es a ner topology than the topology indu ed from
Carp ( ; Rm ) be ause of (3.99). Then the respe tive
R R
R
p m ) is equivalent with the subset DM p ( ; m ) #
onvex  ompa ti ation Y ( ; H R p U % òN DMR; % ( ; m ) of r a( , R m ) dened by (3.51). Let us note that (3.138) is fulp m ) forms a B  oer ive lled, so that the set of all DiPernaMajda measures DM ( ; R p m lo ally ompa t onvex  ompa ti ation of L ( ; ). If 1 p , this onvex  om
R
R
R
pa ti ation is even norm onsistent thanks to Theorem 3.39(iii).
p m m DMR ( ; R ) is lo ally sequentially ompa t provided R R is metrizm m m able. Indeed, if R R is metrizable, so is , R R , and then C ( , R R ) ontains a m
ountable dense subset,76 and therefore the weak* topology of r a ( , R R ) is metrizm able on subsets whi h are bounded with respe t to the dual norm on C ( , RR ) . Moreover,
*
p One an easily see that, for any sequen e { u k }kòN unbounded in L ( ; k"1 , Æ(u k ) onverges weakly* to Æ(0) in Yp ( ; m ).
75
76
R
See, e.g., Bourbaki [144, X.3.3, Theorem 1 and IX.2.8, Proposition 12℄.
Rm
supp(u k ) ¢
) su h that
Ë
196
3 Young Measures and Their Generalizations
# % u pLp Rm guarantees that lr a , Rm i(B % ) is R m ) for every % ò N. However, for every bounded (and thus metrizable) in C ( , R R p ò DM ( ; R m ), there is % ò N large enough for l R r a , Rm i(B % ) to be a (sequeniu
The identity ( )
C ( ,
R
Rm )
*
(
;
)
(
)
*
(
R
)
tially ompa t) neighbourhood of .
Example 3.71 (A renement of DiPernaMajda measures). For a ring G su h that C( ) G L ( ), we an again dene a Ginvariant subspa e H by (3.104) and en
h
dow it by the norm H dened by (3.98). We get thus a lo ally ompa t
R
G #Ö C( ),
p m ). If
onvex  ompa ti ation of L ( ; p m DiPernaMajda measures DM ( ; ). R
R
B  oer ive
it is stri tly ner than the
Example 3.72 (Fonse a's extension of L spa es). For p # 1 one an take H from m m (3.105) and then obtain Y ( ; R ) a onvex  ompa ti ation of L ( ; R ) whi h is H m equivalent with the set F( ; R ) of all Fonse a's measures. It is not B  oer ive, howm " ) with C ( , S m " ) ever. A slight enlargement of H from (3.105) by repla ing C ( , S m yields Y ( ; R ) a B  oer ive, lo ally (sequentially) ompa t onvex  ompa ti aH m m tion of L ( ; R ) equivalent with the set F( ; R ) of Fonse a's measures on ; f. 1
1
1
1
0
1
1
1
also Summary 3.54.
Example 3.73 (Coarser onvex  ompa ti ations).
Let us take
p ò [1; %℄, S #
R
Rm ,
p m ) is equivalent for from (3.107) with the norm dened by (3.108). Then Y ( ; H p ¢ % with L p ( ; m ) endowed by the weak* topology77 while for p # 1
H
1
R
; Rm ) Ê vba( ; Rm ) endowed by the weak* topology, 1 m i.e. the bidual spa e of L ( ; R ). The ane homeomorphism is via the mapping p m H Ù L ( ; R ) adjoint to the operator g ÜÙ g id : L p ( ; Rm ) Ù H . These spa es serve as examples of homogeneous sequentially B  oer ive onvex  omit is equivalent with
*
L
(
*
*
pa ti ations whi h are not
B  oer ive; f. also Examples 2.25 and 2.26. Analogously,
Y H1 ( ; Rm ) Ê r a( ; Rm ) while Y H ( ; Rm ) Ê L p ( ; Rm ) for p ¡ 1. Likewise, the hoi e (3.110) yields Y H1 ( ; Rm ) Ê r a( ; Rm ) and (3.111) yields Y H1 ( ; Rm ) Ê L1%" ( ; Rm ).
the hoi e (3.109) yields
p
Remark 3.74 (Metri ompletion of L p ( ; S)).
d on U # L p ( ; S), the most natural extension is its ompletion with respe t to the metri d , whi h is a omplete metri spa e ( U ; d ) su h that U is embedded densely into U and d U , U # d . p Negle ting the onvex stru ture, Y ( ; S) an a tually be identied with a suitable H p
ompletion of L ( ; S) provided H is separable and satises (3.138). Then an appropriate metri on d an be: d(u1 ; u2 ) # d h (u1 ; u2 ) %
77
For
Having a metri
H2 k #1
"k
d h k (u1 ; u2 ) ; 1 % d h k (u1 ; u2 )
1 p % it is merely the weak topology be ause of the reexivity of L p ( ; Rm ).
3.4 A lass of onvex
where, for
h ò H , d h (u1 ; u2 ) #
P ( h ( x ; u 1 ( x ))
h
 ompa ti ations of
Lp
spa es
Ë 197
" h(x ; u (x))) dx and h is the oer ive 2
H . It is an easy exer ise d if and only if it is bounded and weakly* onvergent when embedded into H via i H . Let us note also p m that d indu es just the strong topology on L ( ; S) provided S # R , p ¡ 1, and H
integrand from (3.138) and the olle tion { k } k #1 is dense in to show78 that a sequen e in
L p ( ; S)
is Cau hy with respe t to *
satises (3.137) and (3.138) with an equality; f. Proposition 3.67(ii).
Summary 3.75.
Some properties of onvex
 ompa ti ations from Examples 3.69
3.73 are summarized in the following table:
Convex  ompa ti ation: L p ( ; Rm ), p ¡ 1 r a( ; Rm ), p # 1 r a( ; Rm ), p # 1 vba( ; Rm ), p # 1 L1%" ( ; Rm ), p # 1 Yp ( ; Rm ), p £ 1 F( ; Rm ), p # 1 p m DMR ( ; R ) DiPernaMajda, rened
B  oer ive
sequentially B  oer ive yes yes yes yes yes no yes yes yes
no no no no no no yes yes yes
Table 3.1. Properties of on rete onvex
Properties: lo ally
ompa t no no no no no no yes yes yes
norm
onsistent no no no no no no no yes yes
linear manifold yes yes yes yes yes no no no no
 ompa ti ations.
p Let us note that, in fa t, the L Young measures
Yp ( ; Rm )
have the worst geomet
ri al/topologi al properties,79 no matter how useful they are and how inspiring role they histori ally played. Let us still add that all onvex and Hausdor. Besides, for DM
p R ( ;
Rm
sures if both
) if the ring
RC
0
R
(
 ompa ti ations in Table 3.1 are homogeneous
F( ; Rm ) is lo ally sequentially ompa t, whi h holds also R is separable and also for the rened DiPernaMajda mea
m ) and
GL
(
) are separable.
Example 3.76 (A norm on Carp ( ; S)). We want to show that, ex ept the ase p # %, p even the whole lo ally onvex spa e Car ( ; S) an be normed in su h a way that (3.93) is satised, although parti ular subspa es may admit stronger norms whi h are some
N
u i }iòN is Cau hy means: :" ¡ 0 ;i0 ò :i1 ; i2 £ i0 : d(u i1 ; u i2 ) ¢ ". For su h d h (u i ; 0)}iòN is bounded in so that, by (3.138), {u i }iòN is bounded in L p ( ; S). The weak* onvergen e of { i H ( u i )}iòN is standard; see, e.g., Bishop and Bridges [121, Se t. 7.6℄, Holmes [392, 78
A sequen e {
R
a sequen e {
Se t. 15℄ or Warga [791, Thm. I.3.11℄.
Rm
m # 1 for simpli ity) is not lo ally ompa t an be shown by taking, 1 * N of Æ(0) ò L w* ( ; r a( )) Ê L ( ; C 0 ( )) , a sequen e { Æ ( k A )}kòN p Y ( ; ) whi h lies in N whenever A has a su iently small positive measure depending on N p ). but not on k ; su h sequen e weakly* onverges but its limit, being zero on A , does not live in Y ( ;
79
The fa t that
Yp ( ;
) (with
for any weak* neighbourhood
R
Note also that this sequen e is not tight.
R
R
R
198
Ë
3 Young Measures and Their Generalizations
times easier to be handled or inevitable for rateoferror estimates, f. the ondition (3.154 ) below. Considering
h Carp
(
p ò [1; %), we put
#
;S)
inf
:(x ; s)ò ,S: h(x ; s)¢a(x)%b s Sp
a L1
(
%b:
)
(3.141)
Carp ( ; S), it will be a norm. The positive homogeneity h Car p S # h Car p S for any ò R is obvious. Let us prove the triangle inequality h % h Carp S ¢ h Car p S % h Carp S . For l # 1 ; 2 and for every " ¡ 0, there are a l " ò L ( ) and b l " ò R su h that h l Carp S £ a l " L 1 % p b l " " " and h l (x ; s) ¢ a l " (x) % b l " s S . Realizing that h (x ; s) % h (x ; s) ¢ [a " % p a " ℄(x) % (b " % b " ) s S , we an estimate
Making possibly a suitable equivalen e on (
;
)
(
1
2
;
1;
2
)
;
(
)
(
;
1
;
h1 %h2 Carp
(
;S)
¢ a
1;
¢ a ¢ h " ¡ 0
;
(
;
)
(
2
)
1;
2;
As
)
;
;
2;
;
1
)
1
;
;
(
%a
"
2;
" L1 ( )
1;
% (b
" L1 ( )
%b
1;
"
%b
2;
")
" % a 2 ; " L 1 ( ) %b 2 ; "
% h
Carp ( ;S)
1
1;
2
Carp ( ;S)
% 2" :
has been arbitrary, the triangle inequality is proved. The fa t that (3.93) is
satised in this ase is plain. The topology generated by the norm (3.141) is ner than
% ò N. Indeed, for every " ¡ 0 there are again a " ò L1 ( ) and b " ò R su h that h Car p ;S £ a " L1 % b " " " p and h ( x ; s ) ¢ a " ( x ) % b " s . Obviously, both a " and b " must be nonnegative. Then S for every % ò N we an estimate
the lo ally onvex topology indu ed by the seminorms  % with
h% ¢
sup
X h ( x ; u ( x )) d x
u L p ;S ¢ % (
)
¢
X a " (x)
u L p ;S ¢ % (
¢ a Passing with
sup
(
)
(
)
% b " u(x) Sp dx
)
" L1 ( )
% b " % p ¢ % p h Car p S % ": (
" to zero, we get the estimate h% ¢ % p h Carp
(
;
(3.142)
)
;S) , whi h shows that the
norm from (3.141) generates the ner80 topology than the natural one. It is ertainly useful to have at our disposal a pro edure how to onstru t larger linear subspa es of
Carp ( ; S) from original ones, without deteriorating signi ant proper
ties of the original spa es. This is handled by the following assertion.81
Proposition 3.77. Let G be a linear subspa e C( ) G L ( ) and H; H ; H be p subspa es of Car ( ; S) equipped with some norms generating ner topologies than the
1
2
In fa t, (3.141) indu es even a stri tly ner topology be ause h Car p ( ;S) ¡ h % # 0 for any h ( x ; s ) # a(x) with a #Ö 0 but P a(x) dx # 0. Anyhow, one an show that h Carp ( ;S) ¢ h% % % h" % with % # 1, h% # max(0; h) and h" # max(0; "h) if one use the al ulations dedu ing the estimate (3.81) from p 1 the boundedness of N h : L ( ; S) Ù L ( ), f. Lu
hetti and Patrone [499, proof of Thm. 3.1(2)℄, uniformly with respe t to h . 81 The reader is en ouraged to prove that similar assertion holds also for the subspa e H 1 H 2 endowed with the norm h H H 1 2 # h H1 % h H2 .
80
3.4 A lass of onvex
(relativized) natural topology on (i)
 ompa ti ations of
Lp
Ë 199
spa es
Carp ( ; S). Then:
H1 % H2 endowed with the norm
The subspa e
h H1 %H2 #
inf
h#h1%h2 h1 òH1 ; h2 òH2
h1 H1 % h2 H2
(3.143)
has a ner topology than the (relativized) natural topology on
Carp ( ; S). Moreover,
H1 and H2 are Ginvariant and satisfy (3.93), then H1 %H2 is Ginvariant and H1 and H2 are separable, then H1 %H2 is separable,
if both
satisfy (3.93), too. Also, if both too. (ii) If
G is a ring (i.e. G  G # G), then the linear hull k H gl l #1
span(G  H) #

h l ; k ò N; g l ò G; h l ò H
(3.144)
Ginvariant linear subspa e of Carp ( ; S) ontaining H . Moreover, if both G and H are separable, then span( G  H ) is separable if equipped with the norm
is the smallest
(3.141). Proof. The fa t that (3.143) denes a norm is quite obvious,82 let us only show the trian
h ; h ò H1 % H2 . By the denition (3.143), for any " ¡ 0 there are h 1 " ; h 1 " ò H 1 and h 2 " ; h 2 " ò H 2 su h that h # h 1 " % h 2 " , h # h 1 " % h 2 " ,
gle inequality. Let us take ;
;
;
and
;
;
;
;
;
h1 " H1 % h2 " H2 " " ¢ h H1 %H2 ¢ h1 " H1 % h2 " H2 ; ;
;
;
(3.145)
;
h 1 " H1 % h 2 " H2 " " ¢ h H1 %H2 ¢ h 1 " H1 % h 2 " H2 : Now we an estimate ;
;
;
;
h % h H1 %H2 ¢ h1 " % h 1 " H1 % h2 " % h 2 " H2 ;
¢ h
1;
# h
1;
As
;
;
;
" H1 %
h 1 " H1
% h2 " H2 % h 2 " H2
" H1 %
h2 " H2
% h 1 " H1 % h 2 " H2
¢ h H1 %H2 % "
% h H1 %H2 % "
:
;
;
;
;
;
;
" ¡ 0 was taken arbitrarily, the triangle inequality for the norm  H1 %H2
has been
proved.
h ¢ C1 % h H1 and h% ¢ C2 % h H2 valid for any % ò N. Taking the de omposition h # h 1 " % h 2 " satisfying (3.145), we an estimate Let us now suppose %
;
;
h% ¢
h1 " % % h2 " % ¢ C1 % h1 " H1 % C2 % h2 " H2 ;
;
¢ max(C with
;
;
1;
;
% ; C 2 ; % ) h 1 ; " H 1
;
% h
;
2;
" H2
;
¢ C % h H1 H2 % "
C % # max(C1 % ; C2 % ). Letting " ÿ 0, we an see that the norm (3.143) generates a Carp ( ; S). ;
;
ner topology than
82
The impli ation
h H1 %H2
# 0 âá h # 0 (in the sense h(x ; ) # 0 for a.a. x ò ) follows from the
assumption that the normed spa es
onvex spa e
H1 and H2 are ontinuously embedded into the Hausdor lo ally
Carp ( ; S); f. Gajewski et al. [342, Chap. I, Rem. 5.13℄.
Ë
200
3 Young Measures and Their Generalizations
The fa t that H 1 % H 2 is G invariant is obvious. Let us now suppose g  h H 1 ¢ C1 g L h H1 and g  h H2 ¢ C2 g L h H2 valid for any g ò G and h belonging to H 1 and H 2 , respe tively. Taking the de omposition h # h 1 " % h 2 " satisfying (3.145),
(
)
(
)
;
;
we an estimate
g  h H1 %H2 ¢ g  h1 " H1 %H2 % g  h2 " H1 %H2 ;
¢ g  h
1;
;
" H1
¢ max(C ; C 1
% g  h 2 )
g L
2;
(
" H2
¢ C g L 1
(
) h1; " H1
) h 1 ; " H 1 % h 2 ; " H 2
% C g L 2
¢ C g L
(
(
) h2; " H2
) h H 1 % H 2 % "
C # max(C1 ; C2 ). Letting " ÿ 0, we obtain (3.93) valid for H1 % H2 . The separability of H 1 % H 2 follows from the separability of H 1 and H 2 by the separate (strong,strong,strong) ontinuity83 of the mapping ( h 1 ; h 2 ) ÜÙ h 1 % h 2 : H 1 , H2 Ù H1 % H2 . The point (i) has been thus proved. p The fa t that span( G  H ) H is a G invariant linear subspa e of Car ( ; S) is obvious. Also, span( G  H ), being a linear hull of the set G  H # { g  h ; g ò G ; h ò H }, is the smallest G invariant subspa e greater than H . Sin e the mapping ( g; h ) ÜÙ g  h : G , H Ù G  H is separately (strong,strong,strong) ontinuous,84 G  H and thus also span(G  H) is separable provided G and H are so. This proved (ii). Å
with
Let us now turn our attention to an important property of generalized Young fun 
pnon on entrating DiPernaMajda measure. In p analogy with this, we will say that a generalized Young fun tional ò Y H % ( ; S) is pnon on entrating if it an be attained by a net {u } ò in L p ( ; S) (in the sense that lim ò i H (u ) # weakly* in H ) su h that the set { u Sp ; ò } is relatively weakly 1
ompa t in L ( ). We saw already one impa t of this property in Proposition 3.43(ii). m Conning ourselves to the ase S # R , let us now state another important onse
tionals. In Se tion 3.2. we dened a
;
*
quen e of this property:
Proposition 3.78 (Youngmeasure representation). (i)
H be separable. Then: p ò YH ( ; Rm ), there exp m ists a (not ne essarily uniquely determined) Young measure ò Y ( ; R ) su h that For any
pnon on entrating
: hòH :
Let
generalized Young fun tional
; h
#
X X
Rm
h(x ; s)
x (d s ) d x :
(3.146)
ò Yp ( ; Rm ) determines by the formula (3.146) a pnon on entrap m ting generalized Young fun tional ò YH ( ; R ).
(ii) Conversely, any
ò Y( ; Rm ) is a onsequen e of the Ball lemma 3.20 if one
Proof. The existen e of realizes that, sin e
H is separable and thus the weak* topology on bounded subsets of
h1;1 % h2 " (h1;2 % h2 ) H1 %H2 # h1;1 " h1;2 H1 %H2 ¢ h1;1 " h1;2 H1 . g  h1 " g  h2 Carp ( ;S) ¢ g L ( ) h1 " h2 Carp ( ;S) and
83
This follows from the estimate
84
This follows from the obvious estimates
g 1  h " g 2  h ¢ g 1 " g 2 L
(
) h Carp ( ;S) .
3.4 A lass of onvex
 ompa ti ations of
Lp
Ë 201
spa es
is weakly* attainable by a bounded sequen e {u k }kòN L p ( ; Rm ) su h that { u k k ò N} is relatively weakly ompa t in L1 ( ), and that also the set 1 { h u k ; k ò N} is relatively weakly ompa t in L ( ) be ause h has at most p growth. p m Then it is also lear that ò Y ( ; R ). Let us go on to (ii). It is lear that (3.146) determines a linear fun tional on H . Let us take a sequen e { u k } k òN su h that, with some % ò R, u k L p ;Rm ¢ % and that generp m ates ò Y ( ; R ) in the sense that lim k Ù P h ( x ; u k ( x )) d x # P P m h ( x ; s ) x (d s ) d x
R 1 m for any h ò L ( ; C 0 (R )). As H is separable, we an even suppose this onvergen e to hold for any h ò H ; f. the proof of Proposition 3.22 and Remark 3.23. Then we an H
*
is metrizable,
p ;
(
)
estimate
!! !!X X !! ! Rm
h(x ; s)
!! ! x (d s ) d x !!! !
!!
!!
¢ sup !!!!X h(x ; u k (x)) dx!!!! k òN !
¢
! !! !! sup !!!!X h(x ; u(x)) dx!!!! ! u L p ;R m ¢ % !
(
where  % is the seminorm of fun tional
# h% ;
)
Carp ( ; Rm ), see (3.82). This shows the ontinuity of the
determined by (3.146); re all that the topology on H
is always supposed
to be ner than the topology indu ed by  % . Altogether, we proved that
u
ò YH ( ; Rm ). p
In view of the biting lemma 1.29, the sequen e { k } k òN an be (if ne essary) modied
u k }kòN again generates and the set { u k p ; k ò N} is L ( ); f. the proof of Proposition 3.81 below. It shows
so that the modied sequen e { relatively weakly ompa t in that
1
Å
is pnon on entrating. The following assertion shows, in parti ular, that every sequen e in
whi h attains a
ò Y Hp ( ;
pnon on entrating  ompa ti ation in
vided the onvex
R
L p ( ; Rm )
m ) does not on entrate energy pro
H
question is ne enough, i.e. if
is large
enough.
Proposition 3.79 (Non on etration of sequen es).85 Let {u k }kòN be a bounded sep m quen e in L ( ; R ) su h that ea h weak* luster point of { i H ( u k )} k òN in H is p nonp
on entrating and let H be su iently ri h,86 e.g. let H ontain H # C ( ) Ô (R ) *
0
85 For
It should be emphasized that this assertion does not hold on oarser onvex
p
p ¡ 1 and Y H ( ;
Rm
)
u k p ; k
0
 ompa ti ation.
Ê L p ( ; Rm ) (i.e. H is from (3.107)) one an take any sequen e {u k }kòN from
Figures 3.779; then obviously
{
i H (u k ) onverges to i H (0) whi h is ertainly pnon on entrating while
ò N} is not relatively weakly ompa t in L ( ). For p # 1 a similar example was already 1
onstru ted for the Fonse a measures in Remark 3.36.
86
H 's. For example, we an ompa tify not the origL p ( ; m ) but the spa e of energies L1 ( ) by taking H0 # C( ) V with V # {v ò m ); ; v ò C ([0 ; %℄) : v ( s ) # v ( s )(1% s p )}, whi h makes Y p ( ; m ) equivalent with 0 0 H0
In fa t, the assertion holds also for a bit smaller
inal spa e
R
Cp (
R
R
DM1R0 ( ; R); H0 ; i) with R0 the smallest omplete subring in C0 (R) and [i(u)℄(x) # u(x)p so that
(
*
we an use Lemma 3.27(ii) on this oarser ompa ti ation.
Ë
202 with
R0
set { u k
3 Young Measures and Their Generalizations
C0 (Rm ) ontaining onstants. Then the k ò N} is relatively weakly ompa t in L1 ( ).
being the smallest omplete subring of
p ;
Proof. Suppose that the assertion does not hold, i.e. { u k weakly ompa t in
L ( ). 1
p;
k ò N} u
is not relatively
Then we an sele t a subsequen e { k } k ò N 1 , every subse
u
quen e of whi h does on entrate energy, i.e. { k
p;
k ò N2 } is not relatively weakly
L ( ) whenever N2 N1 is innite.87 of {i H (u k )}kòN1 and a ner net {i H (u k )} ò onverging to ; p of ourse k ò N 1 for any ò and therefore the set { u k ; ò } is inevitably not 1 relatively weakly ompa t in L ( ). As is also a luster point of { i H ( u k )} k òN , it must be p non on entrating. We have also w*lim ò i H 0 ( u k ) # # H 0 , where is p non on entrating, too. Sin e H 0 is separable, we an now onsider N 1 dire ted by the standard ordering p indu ed from N. By Lemma 3.27(ii), we an see that { u k ; ò } is relatively weakly 1
ompa t in L ( ), a ontradi tion. Å 1
ompa t in
Take a luster point
Let us introdu e another important notion, whi h will serve as a powerful tool
p ; ò YH ( ; Rm ), we say that is a pnon on entrating modi ation of if is pnon on entrating and # holds for any h ò H su h that h(x ; s) ¢ a(x) % o(sp ) with some a ò L1 ( ) and o : R% Ù R satisfying limrÙ o(r)/r # 0. Let us note that the on ept of the p non on entrating modi ation is sensible only if H ontains integrands whi h have (in absolute value) the growth pre isely p be ause otherwise every generalized Young measure is, by the very denition, automati ally the p nonlater. For
on entrating modi ation of itself. Let us also remind that an example of a on rete
ÜÙ was demonstrated in Proposition 3.30 for the
pro edure realizing the mapping
ase of the DiPernaMajda measures. The following assertion justies our notation, showing that mined uniquely by
, if exists, is deter
in question.
Proposition 3.80 (Uniqueness of pnon on entrating modi ation). Every p p ò YH ( ; Rm ) admits at most one pnon on entrating modi ation ò YH ( ; Rm ). p 1 ; 2 ò YH ( ; Rm ) are two pnon on entrating modi ations Let us take h ò H and put h r ( x ; s ) # h ( x ; s ) v r ( s ) with the ut
Proof. Let us suppose
ò YH ( ; Rm ). o fun tion v r given again by (3.38). Without loss of generality we an suppose 1 m that L ( ; C 0 (R )) H . More in detail, if it is not the ase, we an repla e H by 1 H # H % L ( ; C0 (Rm )) and extend ; 1 ; 2 on this enlarged spa e so that again p
of
p Indeed, by DunfordPettis theorem 1.28(ii), { u k ; k ò N} is not uniformly integrable, whi h means ;" ¡ 0 :n ò N ;k n ò N: P xò uk x p ¢n u kn (x)p dx £ ". Putting N # {k n ; n ò N}, we have, for n any N N innite, :K ò R ;n ò N (e.g. n # min(N [K ; %℄)) P xò un x p £K u n (x)p dx £ p p P u n ( x ) d x £ " , whi h shows that the set { u k ; k ò N } is not uniformly integrable. xò u n x p £n
87
{
2
{
;
1
( )
;
( )
}
1
}
2
2
{
2
;
( )
}
3.4 A lass of onvex
p ; 1 ; 2 ò YH ( ; Rm ).
 ompa ti ations of
1
Lp
spa es
Ë 203
2
remain p non
on entrating as well. If one shows 1 # 2 in the sense of H , then it is obvious that it holds for the original fun tionals on H as well. Thus, adopting the agreement 1 m that H ontains L ( ; C 0 (R )), we may and will suppose h r ò H be ause always 1 m h r ò L ( ; C0 (R )). As h r has a growth less than p and both 1 and 2 are p non on entrating modi ations of , we have Besides, the extended fun tionals
and *
1 ; h r
# ; h r # ; h r :
(3.147)
2
Now we want to show that
lim ; h r # ; h :
r Ù
1
(3.148)
1
1 is pnon on entrating, there is a net {u } ò bounded in L p ( ; Rm ) su h that w* lim ò i H (u ) # 1 and the set {u p ; ò } is relatively weakly ompa t in L1 ( ) and
As
therefore, by the DunfordPettis theorem 1.28(ii), this set is also uniformly integrable. This means that, for any
" ¡ 0, one an nd r "
sup ò
As In
X {
x ò ; u ( x )p £ r " }
su iently large so that
u (x)p dx ¢ " :
h ò H Carp ( ; Rm ), we have h(x ; s) ¢ a(x) % bsp for some a ò L1 ( ) and b ò R. parti ular, a is absolutely ontinuous in the sense that, for any " ¡ 0, there is
m " ¡ 0 small enough so that
sup X a(x) dx A measurable A A ¢ m "
¢ ":
x ò ; u (x) £ r} ¢ (C/r)p with C # "1 p 1 p r £ max(Cm " 2 ; r " 2b ) and every ò , we an
Let us noti e that it ertainly holds {
sup ò u L p Rm (
;
).
Then, for every
/
/
/
/
estimate
!!
!!
i H (u ); h r " h # !!!!X h(x ; u (x))(v r (u (x)) " 1) dx!!!! !
!
¢
X {
¢
X {
a ( x ) x ò ; u ( x )£ r }
a(x) dx % X
x ò ; u ( x )£ r }
Passing to the limit with
% bu (x) {
p
dx bu (x)p dx ¢
x ò ; u ( x )£ r }
"
2
%
"
2
# ":
ò , we obtain # . As this holds for any h ò H with the growth less that p , we have shown that is the p non on entrating modi ation of . Å so that we showed that
# < " P ; h > # < ; h " Pd h >. By the estimate d < " d ; h > ¢ H h " Pd h H together with (3.154b) we an see that d Ù weakly*. It remains to prove (3.150). Of ourse, we put again d # P ò K d . Then d Let us go on to (3.149b). For a given
*
*
*
*
" d p H # sup *
h pH ¢1
" d ; h # sup ; h " Pd h
h pH ¢1
¢ sup H h " Pd h H ¢ sup Cd H h pH # Cd H :
*
h pH ¢1
h p H ¢1
*
Statement (iv) follows dire tly from (3.149b) be ause always thanks to (3.152) and (3.153); re all that
*
p
K d YH ( ; S)
B  oer ivity implies losedness due to Propo
Å
sition 2.20(i).
Though the abstra t onstru tion introdu ed above is quite simple, the proper task onsists in a hoi e of
H and a onstru tion of the parti ular proje tors Pd and of
the norms  H and  p H whi h t with a treated on rete problem, an be easily implemented, and satisfy the above required onditions. Let us remark here that (3.154b) is not ne essary for (3.149b) and a tually sometimes (3.149b) must employ another
onstru tion than
d # Pd ; see, e.g., (5.95b). *
P
Obviously there is a great amount of possibilities how to onstru t d , but we mention now only some (hopefully quite representative) examples whi h will be used also in the following hapters.
Ë 209
3.5 Approximation theory
We will use two parameters
d1
and
d2
for dis retisation of
and S, respe tively,
and onstru t our proje tor always as a omposition
Pd $ P where the parti ular proje tors variable
(
d1 ; d2 )
# Pd 1 PdS2 # PdS2 Pd 1 ;
Pd 1 and PdS2
(3.155)
are responsible for the dis retisation in the
x ò and s ò S, respe tively. Possibly either Pd 1 or PdS2
may be the identity.
The following assertion is useful if one wants to verify (3.154 ) for (3.155) from the knowledge of (3.154 ) for the parti ular proje tors
h
Pd
in the form
and
p H and two norms  p H ;H 2 H1 # h pH1 % h pH2 .
suppose that we have given two subspa es p1 The norm  p H 1 p H 2 is dened by p H1 p H2
Pd 1
PdS2 . Let us and  p H2 .
Proposition 3.84. Let there be C , C , C , ; ¡ 0 su h that, for all h ò H and d # d ; d ) ¡ 0, the following estimates hold: 0
(
1
1
2
1
2
2
" " " h " "
"
" Pd 1 h""""H ¢ C d 1 h pH1 1
" " " h " "
and
1
"
" PdS2 h""""H ¢ C d 2 h pH2 ; 2
(3.156a)
2
and
" " " " P h""" p " " d1 " H2
¢ C h pH2 0
" " " " PS h""" p " " d2 " H1
or
¢ C h pH1 :
(3.156b)
0
Then the approximation property (3.154 ) is valid. More spe i ally, for and for
C # max(C1 ; C2 ; C0 C1 ; C0 C2 ), it holds " " " "h
Pd
from (3.155)
" Pd h""""H ¢ C(d 1 % d 2 ) h pH1 pH2 : 1
(3.157)
2
Proof. Let us suppose that, for example, the rst part of (3.156b) is satised. Then we
an estimate:
" " " "h
"
"
"
"
"
"
" Pd h""""H # """"h " PdS2 Pd 1 h""""H ¢ """"h " Pd 1 h""""H % """" Pd 1 h " PdS2 Pd 1 h""""H
¢ C d 1 h pH1 % C d 2 Pd 1 h pH2 ¢ C d 1 h pH1 % C C d 2 h pH2 ¢ (C d 1 % C C d 2 ) h pH1 pH2 ¢ C(d 1 % d 2 ) h pH1 pH2 1
1
with with
2
1
1
0
2
2
2
1
0
1
1
2
2
2
C # max(C1 ; C0 C2 ). If the se ond part of (3.156b) is valid, we get su h estimate C # max(C2 ; C0 C1 ). Å
Remark 3.85 (Approximations of Type III).
Though approximations of Type III may
seem a bit less natural, they are used most often mainly be ause some of them an be implemented by the same way as original, nonrelaxed problems. We have in mind the situation when simply
K d # i H (U d ) ; U d L p ( ; Rm ) nitedimensional:
(3.158)
U d is a onvex subset (or a linear subspa e) of L p ( ; Rm ), but the embedp m ding i H : L ( ; R ) Ù H is not ane provided H ontains at least one nonane integrand, whi h makes eventually K d from (3.158) non onvex.
Typi ally,
*
Ë
210
3 Young Measures and Their Generalizations
U d ontains the L p ( ; Rm ) on a niteelement triangulation Td
1 of like in (3.160) for d # d 1 . For the ase m # 1 and R , the approximation is outlined on Figure 3.11 where an equidistant partition of onto subintervals of the length d is used. Let us illustrate this kind of approximation in the ase that
elementwise onstant fun tions from
S
S
PSfrag repla ements
d
Fig. 3.11:
The onventional non onvex approximation of a Young measure.
We would like to noti e that, in fa t, su h kind of approximation has been already
onstru ted in the Step 2 of the proof of Theorem 3.6. Let us only remark that error estimates an be also obtained for this ase; e.g. if
n # 1 and H # L1 ( ; C(S)) with S Rm ompa t, one an derive90 the estimate
" d C 0 1 ;
[
(
; C ( S )) L
(
¢ Cd 1 2
; C 0 2 ( S ))℄ ;
*
/(
1 2 %1 m%2 )
d # i H (u d ), where u d is pie ewise onstant
R1 onto the subintervals of the length d # d1 .
with
L1
(
; C ( S ))
*
(3.159)
on the equidistant partition of
To ompare (3.159) with (3.169), one should estimate the dimensionality of the resulting problems. For this it is essential that, to over ments of the diameter less than
d1
and
Rn and S Rm by ele
d2 , one needs minimally O(d"1 n ) and O(d"2 m )
mesh points (=variables), respe tively. Therefore, to realize (3.169) the number of mesh points
D
n # 1
as only
must be proportional to
d"1 n d"2 m ,
while for (3.159) we have
D È d "1
is admitted in this ase. In the ase of (3.169), we fa e the question
d1 and d2 to get the highest rate of onverd11 /d22 È onst., whi h yields the rate of error Thus for n # 1 the estimate (3.169) yields the
of an optimal syn hronization between gen e. This optimal ratio is obviously
D as O(D"1 2 1 m%2 n ). " 1 m%2 ) while error O( D 1 2 /(
in terms of rate of
/(
O(D"1 2 /(1 m%2 %1 2 ) ).
)
)
the estimate (3.159) gives a slightly worse rate
Su h omparison with the semidis retisation of Type II, reated e.g. by the proje tor
Pd
*
with
Pd # Pd 1 , is not possible for the ase of (generalized) Young fun tionals
without any spe ial properties. On the other hand, one is often interested only in (generalized) Young fun tionals exhibiting some spe ial properties (like being solutions of optimization or variational problems). Then a semidis retisation of Type II may
90
C in the error estimate depends linearly on the Hölder ';2 , f. [661℄ for details.
We refer to [661, Lemma 3.1℄, realizing that
ontinuity onstants
';1
and
Ë 211
3.5 Approximation theory
appear even far more e ient than the full dis retisation, as we will see in Se tions 4.3.e and 6.6.
An approximation over
3.5.b
The simplest approximation over spa e (or time) is by a dis retisation of
and by
an elementwise homogenization.91 Supposing the reader to be roughly familiar with basi ideas of the niteelement method (FEM), we dis retise the domain
Rn by
a niteelement mesh, say a triangulation. For simpli ity, we will suppose that
is
polyhedral and, for any d 1 ¡ 0, T d 1 is a triangulation of onsisting of elements of
the diameter not ex eeding d 1 . Ea h element E ò T d 1 is therefore a simplex with n % 1
verti es. For d 1 £ d 1 ¡ 0, we suppose that T d 1 Td 1 , this means Td 1 is a renement
* of T . Our aim is to onstru t P so that ( P ) will be elementwise homogeneous d1 d1 d1
generalized Young fun tionals. This will be done ( f. Proposition 3.86 below) if the proje tor
Pd 1
makes spatial averages within ea h element, so that the result
Pd 1 h will
be an elementwise onstant Carathéodory integrand dened by
Pd 1 h(x ; s) # Equivalently, [
1
E
X h( x ; s) d x E
if
x ò E ò Td 1 :
(3.160)
1 Pd 1 h℄(x ; s) # [ Pd 1 h(; s)℄(x) where the average operator P
d : L ( ) Ù
L ( ) is dened by 1
P
d g ( x ) #
1
E
X g( x ) d x E
x ò E ò Td :
if
Let us illustrate the interpretation of the operator ( Young measures, i.e.
#{
91
x } x ò via the mapping
from Lemma 3.4,
*
Indeed, for every
#
p
Pd 1 ) an be identied with an elementwise homogeneous (= onstant d1 x
1
on the ase of the lassi al
H # L ( ; C(S)) and S S ompa t, f. Se tion 3.1. If ò YH ( ; S)
d1
on ea h element) Young measure
* ´( Pd ) ; h µ
*
1
is identied with the Young measure we laim that (
Pd 1 )
(3.161)
#
1
E
#{
X E
x
d1 x } x ò dened by
dx
x ò E ò Td 1 :
if
(3.162)
h ò H # L1 ( ; C(S)), we an write, using Fubini's theorem, that
# ´ ; Pd 1 hµ # 1
1
X h( x ; s) d x H X X E S E E E òTd
1
H X X X h ( x ; s ) x (d s ) d x E E E S E òTd
1
dx
x (d s ) d x
For numeri al approximation of Young measures by an elementwise homogenization see also Pe
dregal [598, 599℄, or also [671℄.
Ë
212
3 Young Measures and Their Generalizations
d1 H X X h ( x ; s ) x (d s ) d x E S E òTd
1
#
where
d1 X h ( x ; s ) x (d s ) d x
S
#X
#
(
d1
)
; h ;
was dened in Lemma 3.4. This proves (3.162).
The elementwise homogenization pro edure is illustrated on Figure 3.12, whi h uses
S
R
1
a onedimensional domain (=an interval) dis retised by an equid1 .
and
distant partition onto the subintervals of the length
S
S (
PSfrag repla ements
Pd 1 )*
d1 Fig. 3.12:
The elementwise homogeneous approximation of a Young measure.
We would like to noti e that su h onstru tion has been already used in the Step 2A of the proof of Theorem 3.6, f. (3.18). Let us investigate some approximation properties of the proposed proje tor (3.160). The requirement (3.151) as well as (3.154a) are ertainly satised for
L ( ; C(S)) 1
h # P supsòS h(x ; s) dx. Also both (3.154b) H # W 1 ( ; C(S)). This follows from the estimates
S % Pd 1 h L 1 ; C S ¢ 2 h L 1 ; C S and h " Pd 1 h L 1 ; C S
with the standard norm H
and (3.154 ) are satised for p
h " P h L 1 ( ; C ( S )) ¢ h L 1 ( ; C ( d1
¢ Cd h W 1 1 C S ;
1
H #
(
;
h " Pd 1 h L1
(
(
)) .
;
))
(
(
))
(
(
))
(
(
))
Then by interpolation92 we obtain
; C ( S ))
¢2
" C d h
1
1
B 1 1 ( ; C ( S )) ;
¢2
" C d h
1
1
W 1 ( ; C ( S )) ;
(3.163)
B pq () denotes a Besov spa e. p Furthermore, let us investigate H # G Ô (R ) from (3.104) used to rene the DiPernaMajda measures. For G # C ( ) ( f. (3.102)) we get the standard DiPernaMajda measures but then Pd H Ö H so that (3.151) is not fullled; to approximate the DiPerna
Majda measures, we would have had to hoose another P d 1 than (3.160), e.g. a on
where
tinuous, elementwise ane interpolation instead of the elementwise onstant averaging as in (3.160). For our hoi e (3.160), the requirement (3.151) will be fullled
G # L ( ), in whi h ase H is not separable, however. Yet there exist subspa es C( ) G L ( ) whi h satisfy (3.151) and yield H separable provided the ring R is separable. For example, we an take for G the spa e
if
G0 #
92
℄ d 1 ¡0
G d1
with
G d1 # g ò L
(
); :E ò Td 1 ; gE ò C(E ) ;
We refer to, e.g., Bergh and Löfström [109℄ for details.
(3.164)
3.5 Approximation theory
or also the losure of
G0 in L
(
). Su h G is separable be ause ea h G d1
and the olle tion of triangulations {T
Ë 213
is separable
d 1 } d 1 ¡0 is supposed ountable. We an easily see
that (3.154a) is fullled, but (3.154b) is not! Nevertheless, (3.149b) an be ensured by another way than via Proposition 3.83, namely by a dire t onstru tion of appropriate
u d1 elementwise onstant on Td 1 su h that i H (u d1 ) onverges weakly* to a given ò p p H on erns, we an take, e.g., p YH ( ; S). As far as the subspa e H H # C0 ( ) V , enm m dowed with the norm h p H # Ô p h C 0 ; C S , where Ô p : Car( ; R ) Ù Car( ; R ) p is dened by [Ô p h ℄( x ; s ) # h ( x ; s )/(1% s ). Then (3.154 ) is satised with C # 1. p p
The very nontrivial fa t that ( P ) : Y ( ; S) Ù Y ( ; S), needed for (3.152), is H H d1 ;
;
(
(
))
*
obvious for the ase of lassi al Young measures93 while for the general ase, this will
be proved later in Proposition 3.86. Let us still investigate some theoreti al properties of the proje tor
P
d from (3.160),
whi h we will be frequently used in what follows. In parti ular, we want to show a quite nontrivial fa t that it satises the hypothesis (3.152).
Proposition 3.86 (Properties of the proje tor Pd ).94 Let a linear subspa e H Carp ( ; S) ontains densely95 some G V with a subspa e G su h that G G L ( ), where G is from (3.164). Then, for every d ¡ 0: p p
(i) ( P ) maps Y ( ; S) into Y ( ; S). H H d p
(ii) For any ò Y ( ; S), ( P ) is elementwise homogeneous, i.e. (1 v ) Ǳ ( P ) is H d d elementwise onstant for any v ò V . p (iii) If ò Y ( ; S) is p non on entrating and v ò V , then H
0
0
*
*
" lim """ (1 d Ù0 "
*
"
v) Ǳ (P d ) " (1 v) Ǳ """"L1 # 0 : *
(
)
R
G V by the norm h GV # Ôp h L ( ,S) ; for S # m it is just
(3.98). Note that P maps G V into G V and is ontinuous with respe t to this norm. d p p lo Sin e we suppose G V H densely, we have Y G V ( ; S) Ê YH ( ; S) so that we an Proof.96 Let us endow
onne ourselves to testintegrands from
G V.
* * * P
d ) : (G V) Ù (G V) p p p * maps Y G V ( ; S) ( G V ) into YG V ( ; S). Let ò YG V ( ; S). By the denition of p YGV ( ; S), there is a bounded net {u } ò L p ( ; S) su h that i GV (u ) Ù weakly* p p * in ( G V ) . As L ( ; S) is dense in L ( ; S) (with respe t to the L norm topology
In view of Theorem 3.66, it su es to show that (
in whi h the embedding
93 94
i : L p ( ; S) Ù ( G V )
*
is ontinuous), we may and will
This follows simply from the expression (3.162) together with Theorem 3.6.
L p Young measures. The onverp ò YH ( ; S) provided v ò V C p (S). p It refers to the natural topology of Car ( ; S) but, of ourse, it su es to have the density in any The point (iii) generalizes the result by Pedregal [598℄ stated for
gen e (iii) holds, in fa t, even for arbitrary
95
ner (e.g. a strong) topology.
96
We use basi ally the te hnique by Kinderlehrer and Pedregal used in [424, 426℄ for the ase of
Young measures. Here it is a bit modied be ause we do not require any non on entration of separability of
H (hen e metrizability of bounded sets in H *).
, nor
Ë
214
3 Young Measures and Their Generalizations
u ò L
; S) although, of ourse, the net {u } ò is generally unbounded ranges the universal index set # N , {nite subset of G V } dire ted by the relation assume that
in
L
(
; S).
(
Without any loss of generality, we an always assume that the index
¢ , , f. also Example 1.4.
Let us now make our onstru tion only for an (arbitrary) element
E ò Td . For every
k ò N, we take a overing (up to a set of zero measure) of E by a ountable (or possibly Pk of pairwise disjoint subsets of the form x kj % " kj E with some x kj ò E and
nite) family
0 " kj ¢ 1/k. The existen e of su h overing follows by the Vitali argument97 [780℄
E has; f. also Figure 6.3 on p. 445. Besides, we an always suppose k%1)th overing is a renement of the kth overing. Then, for # (k ; {h l }),
whatever shape that the ( we put
u (x)
#
. 6 > 6 F
u
0
x"x kj " kj
for
x ò x kj % " kj E ; j ò N ;
elsewhere
Making this onstru tion on every element
(3.165)
:
E ò Td , we get eventually u ò L p ( ; S). As p
u } ò is bounded in L p ( ; S), so is {u } ò, . Sin e YGV % ( ; S) are ompa t for % any % ò R and the universal index set is ri h enough, its image via i must onverge in ( G V ) (possibly only as a ner net but indexed again by , f. Example 1.4) to p
some element in Y G V ( ; S). We want to show that it is just ( P d ) provided the ner {
( ;
)
;
*
*
net is sele ted arefully. First, we take
ò xed. The net {i GV (u )}ò must onverge (possibly as a ner p
ò YGV ( ; S). We want to show that # (P
d ) i G V ( u ). Thus we are to
show # < ; g v > # # d d
for any g ò G and v ò V . Note that P , dened by (3.161), maps G into itself be ause d
we supposed G G 0 with G 0 from (3.164), and that P ( g v ) # ( P g ) v , hen e d d
# . Let us again lo alize our onsidd d erations on E and take g k ò G E pie ewise onstant on the partition P k . Then net) to some
*
*
*
*
X g k ( x ) v ( u ( x )) d x E
whenever
#
x"x kj
H X g k (x)v u k dx k k "j j òN x j % " j E
#
k n k H (" j ) X g k (x j E j òN
% " kj x)v(u (x)) dx #
#
k n k X v ( u ( x )) d x H ( " j ) g k ( x j ) E j òN
#
1
X g k (x) dx X v ( u ( x )) d x E E E
(3.166)
ò is su iently large, namely # (k ; {h l }) with k £ k. Altogether, this
gives
97
k n k H ( " j ) X g k ( x j ) v ( u ( x )) d x E j òN
See also e.g. Dunford and S hwartz [275, Se t. III.12.2℄.
Ë 215
3.5 Approximation theory
i G V ( u ) ; g k
v #
H X g k ( x ) v ( u ( x )) d x E E òT
d
1
#
X g k (x) dx H X v ( u ( x )) d x E E E E òT
#
* ( P d ) i G V ( u ) ; g k
# i GV (u ); P d g k v
d
v #
(1
v) Ǳ (P d ) i GV (u ); g k : *
u ; g k v> Ù #
i
1 v) Ǳ ; g k >. # for every g k ò G pie ewise
Simultaneously, we know that < G V ( ) Therefore, / E .
1 As to (iii), it is an easy exer ise to show that limd Ù0 P g # g strongly in L ( ) for d 1 any g ò L ( ), in parti ular for g # (1 v ) Ǳ , as well. Å
As this holds for an arbitrary ontinuous
g
supported on
*
3.5.
An approximation over S PdS2 in the simplest ase where, instead of a separable m ompa t polyhedral. For every d ¡ 0, take only S R 2
Now we give an example for Bana h spa e
S,
we take
Ë
216
3 Young Measures and Their Generalizations
than
TdS2 . Then denote by PdS2
ea h
TdS2
of S onsisting from elements of the diameter less d2 . For d2 £ d2 ¡ 0, we suppose that TdS2 TdS , this means TdS is a renement of
a niteelement triangulation
2
2
: C(S) Ù C(S) the linear ontinuous proje tor whi h assigns
v ò C(S) the elementwise ane interpolation whi h oin ides with v at all mesh TdS2 . Then we dene PdS2 by
points of the triangulation
PdS2 h(x ; s) # PdS2 (h(x ; ))(s) :
(3.167)
Let us illustrate the interpretation of ( measures, i.e.
H # L ( ; C(S)). 1
PdS2 )
*
again for the ase of the lassi al Young
We will see that this proje tor makes an aggrega
tion of Young measures so that the resulting Young measures are omposed of a 
x ò . The Ld PdS2 C(S) C(S) possesses the base, denoted by {v ld2 }l#12 , su h
nite number of atoms (=Dira measures) at xed supports independent of niteelement subspa e
L d2 l l that ea h v d 2 ò C ( S ) is nonnegative and l #1 v d 2 ( s ) # 1 for any s ò S . Moreover, we L d2 S l l l
an write [ P h ℄( x ; s ) # d2 l #1 h ( x ; s d 2 ) v d 2 ( s ), where s d 2 ò S denotes the mesh points. p If ò Y ( ; S) is identied with the Young measure # { x }xò via the mapping H S * from Lemma 3.4, we laim that ( P ) an be identied with an aggregated Young d2 d2 # { d2 } dened by measure x ò
x d2 x
#
Æ s denotes the L1 ( ; C(S)) we have
where
S ´( Pd
2
*
)
L d2
l H a d (x)Æ s l 2 d2 l #1
with
a ld2 (x) # X v ld2 (s) S
Dira measure supported at
s ò S.
x (d s ) ;
Indeed, for every
(3.168)
hòH #
L d2 l l X X H h ( x ; s d ) v d ( s ) x (d s ) d x 2 2
S l #1 L d2 L d2 l l l l H X h ( x ; s d )X v d ( s ) x (d s ) d x # H X h ( x ; s d ) a d ( x ) d x 2 2 2 2 S l #1
l #1
; hµ # ´ ; PdS2 hµ #
#
d2 X h ( x ; s ) x (d s ) d x
S
#X
#
(
d2
)
; h
d 2 ò Y( ; S ) is taken as in (3.168). Let us note that a l ( x ) £ 0 and L d2 a l ( x ) d2 l #1 d 2 L # PS l#d12 v ld2 (s) x (ds) # PS x (ds) # 1 so that (3.168) a tually determines a Young mea
provided
sure.
a oneS R1 dis retised by an equidistant partition
The aggregation pro edure is illustrated on Figure 3.13, whi h uses dimensional domain (=an interval) and onto the subintervals of the length
d2 .
Ë 217
3.5 Approximation theory
S
S
d2
repla ements (
PdS2 )*
Fig. 3.13:
The aggregation of a Young measure.
We would like to noti e that su h onstru tion has been already used in the Step 2B of the proof of Theorem 3.6.
L1 ( ; C(S)), the requirement (3.151) as well as (3.154a,b) p # L 1 ( ; C 0 ( S )). The is fullled. Besides, also (3.154 ) is satised with C # 1 for H p p S fa t that ( P ) : Y ( ; S) Ù Y ( ; S), needed for (3.152), follows from the obtained H H d2 For the standard norm of
;
*
representation (3.168) together with Theorem 3.6.
PdS2
Of ourse, we ould also think about another onstru tion of
by means of a
suitable higherorder interpolation or another approximation method. When proje tor
S
Rm is unbounded, in parti ular if S # Rm , the onstru tion of the
PdS2 be omes te hni ally more di ult but the previous ideas an be straight
forwardly modied, for example, in the ase of the DiPernaMajda measures using the subring
R
from (3.45), where the metrizable ompa ti ation
homeomorphi with a ompa t polyhedral domain in
Remark 3.87.
R
m.
Rm Ê Rm S m"
1
R
The onstru tion presented here ertainly reminds that one from Se 
Pd : H Ù H determines proje tors FH Ù FH used 2.4 provided Ker Ker( Pd ). If this ondition were supposed, we ould *
tion 2.4. A tually, the proje tors in Se tion
is
*
shorten the proofs of Lemma 3.82 and Proposition 3.83.
Remark 3.88 (Approximations of Type I).
If
S #
Rm and H has the form G V as in
Pd with Pd # Pd 1 from Se t. 3.5.b yields dire tly a full dis retisation (Type I) provided V is nitedimensional as in (3.107), while if V is innitedimensional it yields only a semidis retisation (Type p m p m II). In parti ular, for H from (3.107) where Y ( ; R ) Ê L ( ; R ), this proje tor H
(3.97), (3.101), (3.102), (3.104), or (3.107), then the proje tor
*
makes nothing else than the elementwise onstant approximation of fun tions from
L p ( ; Rm ).
It is easy to see that
PdS2
from (3.167) ommutes with
Pd 1
from (3.160). Therefore,
having a Young measure as in the lefthand part of Figure 3.12 (or 3.13), we an apply the omposed proje tor (
Pd 1 PdS2 )
*
(whi h is the same as (
PdS2 Pd 1 )
*
), whi h gives an
elementwise homogeneous aggregated Young measure. This omposed approximation is outlined on Figure 3.14, f. also Figures 7.1 and 7.3 on pp. 505507.
218
Ë
3 Young Measures and Their Generalizations
S
S
d2
PSfrag repla ements (
Pd 1 PdS2 )*
d1 Fig. 3.14:
The elementwise homogeneous aggregation of a Young measure.
It is lear that this omposed pro edure yields a tually an approximation of Type I, this means onvex and nitedimensional. Also note that (3.156b) is satised. In parti ular, for the ase of the lassi al Young measures with
H # L1 ( ; C(S)) endowed with the standard norm, by Propositions 3.83
and 3.84 one gets the error estimate
valid for
" d W 1 1 ;
[
d $
(
d1 ; d2 )
(
; C ( S )) L 1( ; C 0 2 ( S ))℄ ;
*
¢ Cd 1 %d 2 L1 C S 1
2
L
; ;{a l }l#d2 ; a l £ 0 ; 1
H
L d2 l #1 a l
H
*
U
#
L d2 l #1
dependent on
and, for any
L d2 ò N, dene
# 1;
:h ò H : # X
In their Youngmeasure representations,
s ld2
(3.169)
*
as
or altrnatively also
))
Inspired by Corollary 3.10, one an
u
the nitedimensional onvex subset in
by having
(
*
onsider a xed ountable olle tion { l } l òN dense in
*
;
# [Pd 1 PdS2 ℄ .
Remark 3.89 (Another approximations of Type I).
ò H
(
H
L d2 l #1 a l h ( x ; u l ( x )) d x Ǳ :
's from (3.170) will read as
a l Æ u l ò r a%1 (U). It diers from the
x as s ld2 # u l (x) while a ld2
x
(3.170)
L
# l#d2 a l Æ u l 1
(
x) ,
approximation (3.168)
are now independent of
x. This
sort of approximation is supported by arguments that the set of Young measures is onvex and ompa t, and that ea h element of onvex ompa t sets an be approximated by a onvex ombination of extreme points due to the KrenMilman theorem 1.14, and that the extreme points of the set of the Young measures are Dira measures a.e.,
f. Proposition 3.24. This approximation, devised by V.M. Tikhomirov [769℄ and used e.g. in [39, 40℄ under the name mix of ontrols, does not seem to be indu ed by any proje tor as in Lemma 3.82. Of ourse, ea h su h
is attainable from U ; for this, one
an take the Youngmeasure representation and onstru t a fast os illating sequen e
L d2 Ù , p Y H ( ; Rm ) due to the density of
like in Step 2 of the proof of Theorem 3.6 on p. 128. Moreover, passing the sets (3.170) in rease and their union is dense in {
i H (u l )}lòN . This allows for the onvergen e proof behind this sort of onvex approxi
mation.
3.5 Approximation theory
3.5.d
Ë 219
Higherorder onstru tions by quasiinterpolation
The onstru tions from Se tions 3.5.b and 3.5. falls into more general s heme involv
x or/and in svariables, let us denote it by Cark l p ( ; S) :#
; S %} with k and l referring to order of dierentiability ; ;
ing higher smoothness in
h òCar ( ; S); h Cark l p in x ò and s ò S , respe tively. A natural hoi e seems p
{
; ;
h Cark l p ; ;
(
(
)
:# h Carp
;S)
(
" k " " " h" " " " " k " x " "Carp ( ; S ) " "
% """"
;S)
" l " " " h" " " " " l " s " "Carp"l ( ; S ) " "
% """"
p £ l, with the onvention that, if n ¡ 1, k /x k means all kthorder derival l tives and analogously, if m ¡ 1, / s means all l thorder derivatives. Let us note k l p k 1 ( ; C ( S )) L 1 ( ; C l ( )). We will rely on that, if S bounded, then Car ( ; S) Ê W provided
; ;
;
(3.156a) in ombination with the former property in (3.156b) and present more general
onstru tions devised in [520℄. As for the proje tor
Pd 1 , we may onsider a olle tion of ansatz fun tions {g i }Ii#d11 (
I(d
)
L1 ( ), using also a dual olle tion {g i }i#11 L ( ). Then we dene the operator Pd1 by a quasiinterpolation with respe t to these bases, i.e. *
[
Pd1 h℄(x ; s) :#
I(d1)
i (s)g i (x)
H i #1
)
i (s) :#
with
P
g i (x)h(x ; s) dx *
P
g i (x) dx *
:
(3.171)
The former desired approximation property in (3.156a) will now read as
Pd1 h L1
(
¢ C d k h W k 1
; C ( S ))
1
1
;
(
; C ( S ))
:
(3.172)
I(d1 ) * I(d1) In fa t, this property links the olle tions { i } i #1 and { i } i #1 with ea h other to some
g
g
extent. Sometimes, these olle tions will be orthogonal with respe t to the natural
L2 
type s alar produ t in the sense
X g i (x)g j
*
(
x) dx
Proposition 3.90 (The proje tor Pd 1 ). for
Pd1 *
: [
Moreover,
Pd1 *
olle tions { g i }
Pd1 *
℄x
#
For
I(d1 ) H i #1
#0 ¡0
Pd 1
g i (x) *
i #Ö j for i # j : for
from (3.171), the following formula holds
P
g i ()
P
{
I(d
g i }i#11 *
)
d
g i ()d *
is a Young measure or, in other words,
I(d1 ) i #1 and
(3.173)
:
(3.174)
Pd1 Y( ; S) Y( ; S), provided the *
satisfy
g i £ 0; g i £ 0; i # 1; :::; I(d1 ); *
I(d1) H i #1
g i (x) # 1 *
X g i ( )d
for a.a.
xò
(3.175a)
and
(3.175b)
# X g i ()d ; i # 1; :::; I(d ) : *
1
(3.175 )
220
Ë
3 Young Measures and Their Generalizations
Pd1 *
Proof. The on rete form of adjoint operator
an be obtained straightforwardly
from denitions if one uses the Fubini theorem:
Pd
*
1
g i ()h( ; s)d
I(d1) P
; h # ; Pd1 h # X
*
X H
S i #1
#
I(d1 )
P
1
H i #1 P
g i ()d *
g i ()d *
X X gi
, S
*
I(d1 )
#X Obviously, [
X [ Pd 1 S *
Pd1 *
x (d s )d( x ; )
h( ; s)g i (x)
( )
g i (x) *
X h ( x ; s )¤ H
S i #1 P
x (d s )d x
g i (x)
g i () d *
X g i ( ) d ¥(d s )d x :
℄ x is a positive measure be ause of (3.175a), and moreover
ds) #
I(d1 )
℄x (
H i #1
g i (x) *
P
g i ()X
S
P
(d s )d
g i () d *
#
I(d1) H i #1
g i (x) *
P
P
g i ()d g i () d *
#1
be ause of (3.175b, ). Hen e we proved that this approximation is onformal. For the parti ular onstru tion from Se tion 3.5.b using P0nite elements, the formula (3.171) is the Clément quasiinterpolation [226℄ of 0order and
Pd 1 is indeed a pro
je tor. Now we have still few other options more:
Example 3.91 (Pd 1 by P1/Q1nite elements). Consider a simpli ial mesh and resulting dis retisation Td 1 , and put g i # g i #the hat elementwise ane, ontinuous fun tion orresponding to i th node. Then (3.173) is not satised, but (3.172) with k # 2, *
and (3.175) are satised. The formula (3.171) is the Clément quasiinterpolation of 1order and
Pd 1
is again a proje tor. A dis ontinuous P1variant arises when putting
g i # g i # the ane fun tion supported however only on the parti ular simplex from Td1 , and vanishing at all its nodal points ex ept one. Then we get all properties as in *
the pre eding ontinuous P1 ase. Considering a Cartesian mesh, one an onstru t
gi # gi
*
as tensorial produ t of the P1fun tions, onsidered in 1dimensional vari
ant (possibly ombined in various dire tions). Again, all properties of these examples
Pd 1 . Yet, it should be emphasized that for quadrati (or higherorder) nite elements, (3.172) with k £ 3 an be satised, but (3.175) annot
hold and thus the resulting approximation [ P ℄ Y( ; S ) is not onformal. d1 are inherited by the resulting
*
As to the proje tor
PdS2 , we dene it again as a quasiinterpolation with respe t to
J(d2) some ansatz fun tions { j } j #1
v
C(S) and {v j }Jj#d1 r a(S) Ê C(S) *
(
)
*
1
by the formula
analogous to (3.171), i.e.
[
PdS2 h℄(x ; s) :#
J(d2) H j #1
j (x)v j (s)
with
j (x) :#
P S
h(x ; s)v j (ds) *
P S
v j (ds) *
:
(3.176)
The latter desired approximation property in (3.156a) will now read as
h " PdS2 h L1
(
; C ( S ))
¢ C d l h L1 C l S 2
2
(
;
(
))
(3.177)
3.5 Approximation theory
Ë 221
J ( d 2) * J(d1) and again, in fa t, it links the olle tions { j } j #1 and { j } j #1 with ea h other to some
v
v
extent. Sometimes, these olle tions are so alled Bdual, i.e. orthonormal with respe t to the natural
L2 type s alar produ t in the sense X v i (s)v j S
*
Proposition 3.92 (The proje tor PdS2 ). S * for ( P ) : d2
(
S d2
Moreover, ( P )
*
PdS2 )
*
x
0 1
ds) #
(
For
#
PdS2
for for
i #Ö j; i # j:
(3.178)
from (3.176), the following formula holds
J ( d 2 ) P v ( ) (d ) x S j H v*j : * P v (d ) j #1 S j
(3.179)
Y( ; S) Y( ; S) whenever
v j £ 0; v j £ 0; j # 1; :::; J(d2 ) *
J(d2) H j #1
v j (s) # 1
for all
and
(3.180a)
sòS :
(3.180b)
S * d 2 ) an be obtained straightforwardly from
Proof. Con rete form of adjoint operator ( P
denitions if one again uses the Fubini theorem:
S * ( Pd ) 2
J(d2) P S
; h # ; PdS2 h # X
X H
S j #1
#X
J(d2 )
H
j #1 P S
1
v j (d) *
*
P S
v j (d) *
X h(x ; )v j (s)[v j S,S
*
J(d2) P S
#X
h(x ; )v j (d)
X h ( x ; s )¤ H
S j #1
v j () P S
x (d )
v j (d) *
,
v j (s)
x (d s )d x
x ℄d( ; s )d x
v j (ds)¥ dx ; *
(3.181)
whi h yields the formula (3.179). More pre isely, Fubini's argument holds lassi ally if
x and all
v j , j # 1; :::; J(d2 ), *
are absolutely ontinuous. In the oppo
site ase, one an he k (3.181) by a ontinuous extension of the absolutely on
, 1 ; 2 ) ÜÙ f(s1 ; s2 )2 (ds2 )℄1 (ds1 ) and (1 ; 2 ) ÜÙ PS [PS f(s1 ; s2 )1 (ds1 )℄2 (ds2 ) for any f ò C(S, S). Indeed, onning ourselves (e.g.) on the former ase, for any sequen es 1k Ù 1 and 2k Ù 2 weakly* in r a(S), one has F k (s1 ) :# PS f(s1 ; s2 )2k (ds2 ) Ù P f ( s 1 ; s 2 ) 2 (d s 2 ) #: F ( s 1 ) for any s 1 ò S , hen e also F k Ù F uniformly on S be ause S S is assumed ompa t, and thus eventually PS F k (s1 )1k (ds1 ) Ê Ù Ê P F ( s 1 ) 1 (d s 1 ) # P [P f ( s 1 ; s 2 ) 2 (d s 2 )℄ 1 (d s 1 ). S S S S ℄ x is nonnegative due to (3.180a), and moreover Obviously, [( P ) d2 tinuous ase, relying on the joint (w* w*) ontinuity of the mappings (
P [P S S
*
S * X [( Pd ) 2 S
s ds #
℄x ( )
J ( d 2 ) P v ( ) (d ) x S j * H X v j (d s ) * S P v (d ) j #1 S j
Ë
222
3 Young Measures and Their Generalizations
J(d2 )
#X
H S j #1
x (d )
v j ()
#X
x (d )
S
#1
due to (3.180b). Hen e we proved that this approximation is again onformal. In addition to the aggregation approximation indu ed by (3.167), we have now other options:
Example 3.93 (PdS2 by P1/Q1nite elements). Consider a simpli ial mesh and result¡ 0 a mesh parameter, and put v j #the hat ing dis retisation Td 2 of S with d elementwise ane, ontinuous fun tion orresponding to j th node s j ò S , and v # j Æ s j =Dira 's distribution at s j . Then (3.177) with l # 2, (3.178), and (3.180) are satised, 2
*
PdS2 is a proje tor. Likewise, for a Cartesian mesh, one an onstru t v j as tensorial
and
produ t of the fun tions from Examples 3.6 onsidered in 1dimensional variant, and
vj
*
as Dira 's distributions in parti ular nodal points. Again, all properties of the previ
PdS2 . Let us remark that this operator
ous P1 onstru tion are inherited by the resulting
has been proposed by Tartar [749℄. Let us point out that, for quadrati (or higherorder)
l ¢ 3 an be satised but (3.180) annot hold and thus the PdS2 ) Y( ; S) is not onformal.
nite elements, (3.177) with resulting approximation (
The operators
*
Pd 1 and PdS2 in the form (3.171) and (3.176) always ommute with ea h
other. This follow easily from Fubini's theorem (possibly extended by ontinuity) by the dire t al ulation:
Pd 1 PdS2 h(x ; s) # Pd 1 ( x ; s ) ÜÙ
I ( d 1 ) J ( d 2 ) P g * ( x ) P h ( x ; s ) v * (d s )d x j
i S g i (x)v j (s) H H * * P g i ( x )d x P v j (d s ) i #1 j #1
S
#
I(d1 ) J(d2) P g* ( x )h( x ; s )[L , v*j ℄d( x ;
,S i H H * * P g i ( x )d x P v j (d s ) i #1 j #1
S
# ¨PdS2 ( x ; s ) ÜÙ
3.6
#
where
J ( d 2 ) P h ( x ; ) v * (d ) j S H v j ( s )¡(x ; s) * P v (d ) j #1 S j
s)
g i (x)v j (s)
I ( d 1 ) P h ( ; s ) g * ( )d j
H * P g i ( )d i #1 S
g i ( x )© (x ; s) # [PdS2 Pd 1 h℄(x ; s) ;
L stands for the Lebesgue measure on .
Extensions of Nemytski mappings
An important lass of nonlinear mappings from one Lebesgue spa e into another one is formed by the Nemytski mappings. Here we want to study a (perhaps somewhat surprising) feature that they may admit an ane ontinuous extension on appropriate onvex
 ompa ti ations. Let us realize that these mappings are (ex ept trivial
Ë 223
3.6 Extensions of Nemytski mappings
ases) nonlinear but with respe t to the original linear stru ture of Lebesgue spa es whi h may be (and mostly is) deformed. This deformation makes possible that the extended mappings are ane in some (or all) arguments. Let us begin with the Nemytski mappings
3.6.a
N'
of one argument only.
Oneargument mappings: ane extensions p; q ò [1; %), two separable Bana h spa es S1 and S2 ,
We will onsider
U 2 # L q ( ; S2 )
U 1 # L p ( ; S1 ) ;
endowed respe tively by the norm bornologies tion 2.5) and a Carathéodory mapping
B1
B2
and
(3.182)
( f. the notation from Se 
' : , S1 Ù S2 satisfying the growth ondition
;a ' ò L q ( ) ;b ' ò R% :(x ; s) ò , S : 1
p/ q
'(x ; s) S2 ¢ a ' (x) % b ' s S1 :
Re all that (3.183) guarantees the Nemytski mapping
(3.183)
N ' : L p ( ; S1 ) Ù L q ( ; S2 )
dened by
N' (u)℄(x) # '(x ; u(x))
(3.184)
[
to be ontinuous and bounded; see Theorem 1.24. For
h òCarq ( ; S2 ), we dene S ' h by [
In other words, For any
S'
B ò B1 ,
S ' h℄(x ; s) # h(x ; '(x ; s)) :
substitutes the fun tion
'
into the Carathéodory integrand
S hB ¢ hN'
we have the estimate '
(3.185)
(
h.
B ) ; it holds even as equality. q
Car ( ; S2 ) ' fulls (3.183) whi h ensures N' (B) ò B2 for every B ò B1 . p Again, we dene here two linear homeomorphi al embeddings 1 : Car ( ; S1 ) Ù q CB1 (U1 ) and 2 : Car ( ; S2 ) Ù CB2 (U2 ) by [ l h℄(u) # P h(x ; u(x)) dx with l # 1; 2. Furthermore, let us onsider two linear subspa es H1 Carp ( ; S1 ) and H2 Carq ( ; S2 ). Of ourse, FH l will mean l (H l ) % { onstants on U l }, l # 1; 2. It turns out
Therefore, it is easy to see that into
Carp ( ; S
S'
is a linear ontinuous mapping from
1 ) provided
that the ondition
S ' (H2 ) H1 is essential98 for
N'
to admit an ane ontinuous extension
(3.186)
N ' :
M(
F1 B1 )
Ù
F2 B2 ).
M(
p
 ompa ti ation Y H1 ( ; S1 ) is B1  oer ive (in parti ular if (3.138) is satS ' (H2 ) H 1 # lCarp ( ;S1 ) H1 p p be ause eventually Y ( ; S1 ) Ê Y ( ; S1 ) by Theorem 3.66 and by Proposition 2.20. On the other H1 H 1 p hand, a pre ise knowledge of the losure of H 1 in the natural topology of Car ( ; S1 ) is usually not at 98
In fa t, if the onvex
ised), then the ondition (3.186) an be weakened by requiring only
our disposal in on rete ases.
224
Ë
3 Young Measures and Their Generalizations
Proposition 3.94 (Ane extensions of Nemytski mappings). N'
be valid. Then the Nemytski mapping
p
q
Let (3.183) and (3.186)
admits an ane ontinuous extension
p
YH1 ( ; S1 ) Ù YH2 ( ; S2 ). This extension oin ides on YH1 ( ; S1 ) with the adjoint operator S ' : H 1 Ù H 2 . Alternatively, we an say that N ' possesses a ontinuous ane *
*
*
extension M(F1 B1 ) ping to
Ù M(F
2
B2 ) whi h oin ides on M(F1 B1 ) with the adjoint map
f2 ÜÙ f2 N' : FH2 Ù FH1 . Altogether, the following diagram ommutes: S' *
p YH1 ( ; S1 )
✲ Y q ( ; S2 ) H2
✒ ■ i H1 ✻❅ i H2 ✻ N' ❅ q p 1 2 L ( ; S1 ) ✲ L ( ; S2 ) e H2 ❅ ❄ ✠ e H1 ❘ ❄ ❅ N ' ✲ M(F2 B2 ) M(F1 B1 ) *
*
# N' . Then (3.183) implies bounded f2 ò FH1 provided f2 ò FH2 be ause of the obvious identity q ( 2 h ) N ' # 1 ( S ' h ) valid for any h ò Car ( ; S2 ). Therefore, by Proposition 2.32, the ontinuous ane extension # N ' : M(F1 B1 ) Ù M(F2 B2 ) does exist and
oin ide with the adjoint mapping to Q : F H 2 Ù F H 1 : f ÜÙ f N ' . On the other hand, it is obvious that S ' : H 1 Ù H 2 is ontinuous and S ' i H 1 # i H2 N' be ause of the identity Proof. Let us employ Proposition 2.32 with and (3.186) implies
*
S ' *
i H 1 ( u ) ; h # i H 1 ( u ) ; S ' h #
*
*
X h ( x ; ' ( x ; u ( x ))) d x
# i H2 N' (u); h
u ò U1 . Therefore, the restri tion of S ' the ane ontinuous extension of N ' .
valid for any
h ò H2
*
*
and
on
p
YH1 ( ; S1 ) realizes
Å
By applying Theorem 3.66 twi e, we obtain eventually the above diagram.
It is now lear that various hoi es of
H1 and H2 give various on rete representa' when H1 and H2 are onsid
tion of (3.186), whi h is, in fa t, a ertain ondition on
ered as xed. We will mention only a few examples. For some of them it will be very di ult to hara terize (3.186) in terms of
' pre isely, so that mostly we will be able
to pose only su ient onditions.
Example 3.95 (The largest H makes (3.186) void). Let us take H # Carp ( ; S ) and H Carq ( ; S ) arbitrary and suppose (3.183). Then (3.186) is always fullled. Indeed, q q any h ò Car ( ; S ) satises h ( x ; s ) ¢ a h 2 ( x ) % b h 2 s S2 with a h 2 ò L ( ), whi h 1
2
1
1
2
2
2
1
2
enables us to estimate
!! ! !![ S ' ( h 2 )℄( x ; s )!!!
¢ a h2 (x) % b h2 '(x ; s)q ¢ a h2 (x) % b h2 a ' (x) % b ' s Sp 1q /
¢ a h2 (x) % 2q" b h2 a q' (x) % 2q" b h2 b q' s Sp 1 : 1
1
q
Ë 225
3.6 Extensions of Nemytski mappings
In view of (3.183),
q
a h2 % 2q"1 b h2 a ' ò L1 ( ), and therefore S ' (h2 ) ò Carp ( ; S1 ) # H1 ,
hen e (3.186) is a tually satised. Let us observe that the weakest mode of the ondition (3.183) is for
q # 1.
Example 3.96 (The substitution h Ǳ ). Let us put S # R, q # 1, and H # G R p with some subspa e C ( ) G L ( ). Then (3.183) means pre isely ' òCar ( ; S ). Taking some linear subspa e H Carp ( ; S ), one an easily verify that (3.186) is valid for any ' ò H if and only if H is G invariant; see (3.91). If H is G invariant, p we an extend the Nemytski mapping N h : L ( ; S ) Ù L ( ) with h ò H arbitrary, 2
*
2
1
1
1
1
1
1
1
1
1
obtaining the identity
:h ò H : ò YHp1 ( ; S ) : 1
when one identies99
H2
*
with
Sh # h Ǳ *
1
G
*
; for
G#L
(
(3.187)
) or G # C( ) f. Example 3.50. Also
note that we did not use here the ondition (3.93) but, on the other hand, we have got only a ontinuous mapping
p
ÜÙ h Ǳ : YH1 ( ; S1 ) Ù G
*
with
h
xed, whi h is a
weaker result than that obtained in Proposition 3.43. Eventually, note that for
q ¡ 1,
the ondition (3.183) oin ides with (3.94). This allows us to extend Nemytski map
Nh : L p ( ; S1 ) Ù L q ( ).
pings
Example 3.97 (Mappings between DiPernaMajda measures). Let us take H # C( ) p C (S )) and H # C ( ) Ô q (R ) with R arbitrary omplete subring of C (S ); f. 1
Ô (
0
1
0
2
2
S1 and S2 are nitedimensional. We will onsider an autonomous ase, i.e. a Nemytski mapping N ' with ' ( x ; s ) # # ( s ) for some # : S1 Ù S2 ontinuous and satisfying the growth ondition
Example 3.47 for the ase when
Then
N'
p/ q
#(s) S2 # O( s S1
)
for
s
S
1
Ù :
(3.188)
 omh2 ò H2 in the form g l ò C( ), and v l ò C0 (S2 ), we have
possesses an ane ontinuous extension on the respe tive onvex
pa ti ations. Indeed, (3.186) is fullled be ause, for every
h2 (x ; s) #
L l #1
q
g l (x)v l (s)(1 % s S2 ) with s ò S2 ,
obviously [
S ' h℄(x ; s) #
# q
L H g l ( x ) v l ( # ( s ))(1 l #1
% #(s) Sq 2 )
L 1 % #(s) Sq 2 (1 H g l ( x ) ¬ v l ( # ( s )) 1 % s Sp 1 l #1
% s Sp 1 ):
p
v l (#(s))(1 % #(s) S2 )(1 % s S1 )"1 ò C0 (S1 ) provided # satises the growth ondition (3.188), we have surely S ' h ò H 1 , hen e (3.186) is a tually valid. In parti ular, if S1 and S2 are nitedimensional, we an substitute any DiPernaMajda measure p from DM 0 C S1 ( ; S1 ) into the fun tion # with the growth (3.188), the result being some q DiPernaMajda measure from DM ( ; S2 ). R
As
(
99
)
In terms of (3.185),
S h : G Ù H1 in (3.187) is dened as g
ÜÙ g  h with [g  h℄(x ; s) # g(x)h(x ; s).
226
Ë
3 Young Measures and Their Generalizations
Example 3.98 (Mappings from Young to DiPernaMajda measures). Let us take H as in Example 3.97 and H # C( ) C p (S ); f. Example 3.46 for the ase when S is nitedimensional. We will onsider the Nemytski mapping N ' with ' ( x ; s ) # # ( s ) for some # : S Ù S ontinuous and satisfying the growth ondition 2
1
1
N'
1
2
Then
1
p/ q
#(s) S2 # o s S1
for
s
S1
Ù :
(3.189)
possesses an ane ontinuous extension on the respe tive onvex
 om
pa ti ations. Indeed, (3.186) an be veried analogously as in the pre eding example
p
q
v l (#(s))(1 % #(s) S2 )(1 % s S1 )"1 ò C0 (S1 ). Let us note that, for S1 # S2 # m R and p # q, this Nemytski mapping operates from the Young measures Yp ( ; S1 ) p into the DiPernaMajda measures DM ( ; S2 ), although the former one is a stri tly R
oarser onvex  ompa ti ation than the later one. Of ourse, this is possible thanks to the growth restri tion (3.189). Likewise, if p ¡ q and ' ( x ; s ) # # ( s ) # s , then S ' q p m m embeds Y ( ; R ) into DM ( ; R ); for p # q f. also Remark 3.29. R be ause now
*
Example 3.99 (Canoni al surje tion). A very spe ial ase appears for S # S and '(x ; s) # s, whi h fulls (3.183) with p # q. Then N' is just the identity on L p ( ; S ), and (3.186) means pre isely H H . The ontinuous ane extension of N' (if 1
2
1
2
1
it exists) is just the anoni al surje tion from the ner onvex
p
p
YH1 ( ; S1 ) onto the oarser onvex  ompa ti ation YH2 ( ; S1 ).
3.6.b
 ompa ti ation
Twoargument mappings: semiane extensions
Let us pro eed this se tion to some universal and often used assertions about spe ial extensions of the twoargument Nemytski mapping
L ( ; S
3 ),
dened by
N ' : L q ( ; S1 ) , L p ( ; S2 ) Ù
N ' (y; u) (x) # '(x ; y(x); u(x)) ;
' : ,S1 ,S2 Ù S3 a Carathéodory mapping, S1 ; S2 ; S3 separable Bana h spa es, q; p; ò [1; %℄. Let us agree that, for notational simpli ity, we will write likewise in the nitedimensional ase s  s in pla e of < s ; s > with s ò Si and s ò Si . We will start with an extension only in the se ond argument, the rst
with
S3
being reexive, and
*
*
*
*
argument remaining in the original Lebesgue spa e. As this extension will be ane only in the se ond argument, we will speak about a semiane extension. Parallel to Corollary 2.33 together with Proposition 3.94, we now have:
Lemma 3.100 (Semiane extensions of Nemytski mappings). Let q; p ò [1; %), ò [1 ; %), let C ( )invariant linear spa es H Carp ( ; S ) and H Car ( ; S ) be given, and, for ' y dened by [ ' y ℄( x ; s ) # ' ( x ; y ( x ) ; s ), let 2
S 'y (H3 ) H2
and
y ÜÙ S 'y : L q ( ; S1 ) Ù Carp ( ; S2 )
with respe t to the norm on
'.
2
Carp ( ; S
Then the Nemytski mapping
N'
2 ),
with
3
be ontinuous
3
(3.190)
S 'y dened by (3.185) with 'y instead of ' u)℄ (x) # '(x ; y(x); u(x)) admits
dened by [N ( y;
Ë 227
3.6 Extensions of Nemytski mappings
an extension
'
p
N : L q ( ; S1 ) , YH2 ( ; S2 ) Ù YH3 ( ; S3 ) dened by
'
N (y; ) # S*'y :
(3.191)
If restri ted on
,
p
L q ( ; S1 ) , YH
;
% ( ; S2 ) with '
(strong weak*,weak) ontinuous and
% ò
R%
arbitrary, this extension is
N (y; ) is ane.
y, we use Proposition 3.94 when realizing that S 'y (H3 ) H2 is just S 'y i H2 (u) # N'y (u) # N ' (y; u). For nets Ù weakly* in H2 and for y Ù y strongly in L q ( ; S1 ), we have Proof. For xed
(3.186) up to notational modi ations. In parti ular we have
*
*
¼N
'
(
'
y ; )"N (y; ); h½ # ´S 'y " S 'y ; hµ *
*
# ´ " ; S 'y hµ % ´ ; (S 'y "S 'y )hµ Ù 0 for any
h ò H3 . Here we used the ontinuity of y ÜÙ S 'y assumed in (3.190).
We will o
asionally write g  ( ' y ) instead of < g; ' y >, meaning a fun tion
,S2 Ù R : (x ; s) ÜÙ means < ; g  ( ' y )>.
100
Of ourse, we use here an
with
228
Ë
3 Young Measures and Their Generalizations
H is C( )invariant, (3.192a) ensures101 that ' y Ǳ òr a( ; S3 ). By (3.192b), q p ' y℄(x ; s) S3 ¢ a(x) % 1 s S2 with a # a1 % b1 y S1 ò L ( ), whi h yields ' y Ǳ ò L ( ; S3 ) by Proposition 3.43(iii), modied for the S3 valued '
ase. Obviously, for # i H ( u ), we get ' y Ǳ # N ' y ( u ) # N ( y; u ) so that (3.193) ' a tually determines the extension of the original mapping N . Proof. As
/
/
we have ensured [
It remains to show the ontinuity of the extended mapping. Let us take a net
y in the strong topology of L q ( ; S1 ) (then su h net must be p % q eventually bounded in L ( ; S1 )), and a net { } ò Y H % ( ; S2 ) with some % ò R
onverging weakly* in H to ; we an use the ommon dire ted index set without any loss of generality. We want to show that ' y Ǳ Ù ' y Ǳ weakly in L ( ; S3 ). By (3.192b), we an see that the net { ' y Ǳ } ò is eventually bounded in L ( ; S3 ) and therefore it su es to show ' y Ǳ Ù ' y Ǳ weakly* in r a( ; S3 ). For every g ò C( ; S3 ) we an write {
y } ò
onverging to
;
*
*
'
y Ǳ " ' y Ǳ ; g # ' y Ǳ ( " ) ; g % (' y " ' y) Ǳ ; g #: I % I : (1)
(2)
I # " ; g  (' y) , so that I Ù 0 be ause Ù weakly* in H and be ause g  (' y) ò H by (3.192a).
p
Let h # Ô (1), whi h means h ( x ; s ) # s . Obviously, h p q òCar ( ; S2 ). MoreS2 over, we may suppose that H ontains h p q so that h p q Ǳ has a good sense. If possibly h p q ò Ö H , we an repla e H (just for the purpose of this proof ) by H % H p q where H p q # L q ( )  {h p q } òCarp ( ; S2 ). Iterating this tri k, we may and will also suppose 1 that L ( ) 1 ò H . By the ontinuity arguments, the following three general relations p are at our disposal for any ò Y ( ; S2 ): H
As for the rst term, we have
(1)
(1)
*
/
/
/
/
;
;
/
:h ; h ò H : h ¢ h âá h Ǳ ¢ h Ǳ ; :g ò L ( ) : (g 1) Ǳ # g ; 1
2
1
2
1
(3.194a)
2
1
:h ò H; h £ 0 :
; h
#
X
(3.194b)
h Ǳ dx # h Ǳ r a : (
)
(3.194 )
Then the se ond term an be estimated by means of (3.192 ) and the Hölder inequality as:
I # ; g  (' y " ' y) (2)
¢ ´ ; g S3 (a 1 % b y Sq"1 1 % b y Sq"1 1 % h p q ) y " y S1 µ *
¢
101
" " " " " g S3 ( a 2 " *
1
1
2
2
2
/
"
% b y Sq"1 % b y Sq"1 % h p q Ǳ ) y " y S1 """"" 1
2
1
2
In fa t, we suppose here, for a moment, that
fullled if
2
H is normed appropriately.
2
/
g  (' y) H
¢ C y g C S 3 (
;
*
)
r a( )
, whi h an be always
Ë 229
3.6 Extensions of Nemytski mappings
"
"
¢ g C S """" y " y""""L q S1 """"a % b y Sq"1 % b y Sq"1 % h p q Ǳ """" q " "L 3 ;
(
Therefore
*
;
(
)
1
2
)
1
2
2
2
/
(
)
:
I Ù 0 as well.
Å
(2)
For the ane extension of a given mapping, the spa e of test fun tions
H1
is to
be su iently large. On the other hand, for the sequentially on ept whi h is onventional in omparison with the on ept of nets,
H1
should be separable. The proof of
the following assertion will also be an interesting illustration of the usage of the norm (3.141) in addition to the proof of Proposition 3.77(ii).
Proposition 3.102 (Separability of H ).
Let
(3.192b, ) is fullled. Then the linear
subspa e
H # span g  ('y); g ò C( ; S3 ); y ò L q ( ; S1 ) *
(3.195)
Carp ( ; S ) is separable with respe t to the norm (3.141), and thus in the natural topolp ogy of Car ( ; S ), too. of
2
2
Proof. Let us prove the ontinuity of the mapping
y ÜÙ ' y : L q ( ; S1 ) Ù Carp ( ; S2 )
with respe t to the norm (3.141). By (3.192 ) and by Young's inequality, we have
'(x ; y1 (x); s) " '(x ; y2 (x); s) S3
¢ a(x) % b y (x) Sq"1 % b y (x) Sq"1 % s Sp 2q y (x) " y (x) S1 1
1
1
¢ a % b y
1
/
2
q "1 S1
% b y
1
q "1 S1
y 1
2
"y %
Æ q q
2
2
"q q p Æ s % y1 "y2 S1 1
q
# a Æ (x)%b Æ s Sp 2
q "1
q "1
q
Æ ¡ 0 provided a Æ :# (a % b y1 S1 % b y2 S1 ) y1 " y2 S1 % Æ1"q y1 " y2 S1 /q q and b Æ :# Æ ( q "1) / q . By (3.141), we have for any " ¡ 0 for any
" " " " 'y1
" 'y
2
" " " "Carp
#
inf
p
a L1
a ( x )% b s S2 £ ' ( x ; y 1 ; s )" ' ( x ; y 2 ; s ) S3
(
)
% b ¢ inf a Æ L1 % b Æ ¢ (
Æ ¡0
)
"
2
"
%
2
Æ ¡ 0 is hosen so small that b Æ ¢ "/2 and then y1 " y2 L q ;S1 is so small ¢ "/2; note that a Æ L1 # O((1 % Æ1"q ) y1 " y2 qL q ;S1 ). This shows p q that the mapping y ÜÙ ' y is even uniformly ontinuous from L ( ; S1 ) to Car ( ; S2 ). Moreover, we an dedu e that also the mapping ( g; y ) ÜÙ g  ( ' y ) : C ( ; S3 ) , L q ( ; S1 ) Ù Carp ( ; S2 ) is ontinuous (even uniformly on bounded sets) be ause of provided
)
(
a
that Æ L 1 ( )
(
)
(
)
*
the obvious estimate
" " " " g 1 ( ' y 1 )
" g (' y 2
" " " "Carp
2)
¢ """"(g "g )  (' y )""""Carp % """" g  (' y " ' y )""""Carp ¢ """"g "g """" C S """" 'y """"Carp % g C """"'y "'y """"Carp : 1
1
As both
2
2
1
(
;
*
3)
2
1
1
2
(
)
2
1
2
C( ; S3 ) and L q ( ; S1 ) are separable, the spa e (3.195) is also separable *
if equipped with the norm (3.141).
Ë
230
3 Young Measures and Their Generalizations
For 1storder optimality onditions, we will also need a smoothness property of the extended Nemytski mapping
H
pose that
is normed so that
H
N
'
. We will use Convention 1.55. Also we will sup
*
H
is a Bana h spa e; re all that
always admits a
p
norm generating a topology ner than the natural topology oming from Car (
; S2 );
f. the universal norm from Example 3.76.
Lemma 3.103 (Dierentiability of semiane extensions).102 Let H be a C( )invariant p linear subspa e of Car ( ; S ), q ò [2 ; %), p ò [1 ; %), ò (1 ; %), let ' satisfy
2
(3.192b) and
:g ò L ( ; S
and let
*
3
:y ò L q ( ; S ) :
)
g  (' y) ò H ;
1
(3.196)
'(x ; ; s) : S1 Ù S3 be ontinuously dierentiable su h that
:g ò L ( ; S ) :y; y ò L q ( ; S ) :
g  (' r y)  y ò H ;
*
3
;a ò L q
/(
3
q")
(
/(
4
q "2 )
' r (x ; r; s) L S1
¢ a (x) % b r Sq1" % s Sp 2q" (
(
(
;
S3 )
3
) ;b4 ; 4 ò R% : q "2 )/ b4 r1 S1 %
¢ a (x) %
(
4
(3.197a)
) ;b3 ; 3 ò R% :
;a ò L q
1
(
)/
)/
q
3
3
' r (x ; r1 ; s) " ' r (x ; r2 ; s) L S1
;
(3.197b)
(
q "2 )/ b4 r2 S1 % (
;
S3 )
p ( q "2 )/ q
4 s S2 r 1 " r 2 S1 :
(3.197 )
'
: L q ( ; S ) , Y Hp ( ; S ) Ù L ( ; S ) is sepap q rately103 Fré het dierentiable at any ( y; ) ò L ( ; S ) , Y H ( ; S ) with the dierential ' q N ( y; ) ò L( L ( ; S ) , H ; L ( ; S )) given by
Then the extended mapping
N
1
2
3
1
*
1
N
'
'
(
2
3
y; ) ( y ; ) # (' r y Ǳ )  y % ' y Ǳ
(3.198)
) : H Ù L ( ; S3 ) is (weak*,weak) ontinuous. Moreover, y N ' (; ) : L q ( ; S1 ) Ù L(L q ( ; S1 ); L ( ; S3 )) is lo ally Lips hitz ontinuous uniformly with re' p q % spe t to ò YH % ( ; S2 ) for any % ò R and [ N ( ; )℄( ) : L ( ; S1 ) , H Ù L ( ; S3 ) p % is lo ally Lips hitz ontinuous uniformly with respe t to ò YH % ( ; S2 ) for any % ò R . and N ( y;
*
*
;
;
Furthermore, if
sup
g L ;S 3
*
(
)
¢1
sup
g  (' y1 "' y2 ) H ¢ C y1 L q
y L q ;S1 ¢1 g L ;S3 ¢1
(
102
If
(
% y
;S1 )
L q ( ;S1 ) y 1 "y 2 L q ( ;S1 )
g  (' r y)  y H ¢ C y L q
(
;S1 )
;
(3.199a) (3.199b)
)
*
)
' does not depend on s, i.e. '(x ; r; s)
tiability properties of the Nemytski mapping also Krasnoselski at al. [442℄.
103
2
(
'
This means that both
N (y; ) and N
' (
# #(x ; r), then Lemma 3.103 speaks about the dierenN# : L q ( ; S1 ) Ù L ( ; S3 ). For su h sort of results see
; ) are Fré het dierentiable.
3.6 Extensions of Nemytski mappings
R% Ù R% ontinuous in reasing, then
'
Ë 231
p
N : L q ( ; S1 ) , Y H ( ; S2 ) Ù q * L(L ( ; S1 ) , H ; L ( ; S3 )) is lo ally Lips hitz ontinuous. Finally, if # 1, all these * * statements remain valid with L ( ; S3 ) and L ( ; S3 ) repla ed respe tively by C ( ; S3 ) and r a( ; S3 ) as far as the partial dierential N on erns.104 with some
C:
'
)℄( ) # ' y Ǳ is obvious be ause ' N (y; ) is linear. Also it is obviously the Fré het dierential. Besides, the mapping ÜÙ ' y Ǳ is (weak*,weak) ontinuous be ause, for Ù weakly* in H * and for * every g ò L ( ; S3 ), we have Proof. The expression for the omponent [ N ( y;
'
y Ǳ ; g # ; g  (' y) Ù ; g  (' y) # ' y Ǳ ; g ;
where also (3.196) has been used.
'
y; ). Let us note that, by (3.192b) and p (3.197b), both ' y Ǳ and (( ' r y ) y ) Ǳ live in L ( ; S3 ) for any ò Y ( ; S2 ); f. PropoH sition 3.43(iii) modied for the S3 valued ase. First, let us noti e that, by (3.197 ), we Let us al ulate the omponent y N (
have
" " ' " " " " " " "
¢
(y% " y ) " ' y
"
1 " "" X "' " 0 "" r "
" "S3
0
;
(
% b y Sq1"
2
(
)/
4
2
(
4
" "S3
% b y % " y Sq1"
2
(
)/
4
)
4
¢ 2" "(a (x) % b y Sq1" 1
" "
(y% " y )
 y d " " (' r y)  y """"
"
(y % " y ) " ' r y""""L S1 S3 y S1 d "
" X (a4 (x)
for any
" " 1 " " " X ' r " " " "" 0
1
¢
" " "
" (' r y)  y """" #
% b y Sq1" (
4
% s Sp 2q"
)/
(
2
)/
)/ q
% s Sp 2q" (
2
)/ q
4
0 ¢ " ¢ 1, where b # b (1 % max(1; 2q
4
4
2
4
/
)
)
" y S1 y S1 d "
y 2S1 :
"1 )). Using (3.194) together with
the onvention from the proof of Lemma 3.101 (this means here that we an suppose
h p q"2 (
" " ' " " " " " " "
¢
)/
q ò H ), one an estimate
(y % " y ) " ' y
"
"
2
¢
"
2
" "L ( ;S3 )
" " " " "( a 4 "
% b y Sq1"
" " " a4 " " "
% b y Sq1"
" " "
Ǳ " ((' r y)  y ) Ǳ """"
2
(
)/
2
(
% b y Sq1"
)/
4
(
4
4
2
(
% h p q" 4
% b y Sq1" 4
)/
2
)/
2
(
% h p q" 4
" 2 " " )/ q Ǳ ) y S1 " " ( ;S ) "L 3
2
(
" " " " "2 q  " " " y" )/ q Ǳ " " "L q q"2 ( ) " " L ( ;S1 )
/(
)
" Ù 0 thanks to the assumption (3.197 ). By the denition of q the dierential, for any y ò L ( ; S1 ), one gets
whi h tends to zero for
'
# 1, the partial dierential y N remains valued in L ( ; S ) be ause ' ) # (' r y Ǳ )  always q £ 2 ¡ 1; indeed, y N ( y; ) ( y; y ò L ( ; S ) be ause y ò L q ( ; S ) and
104
Let us note that, for
1
1
'r y Ǳ ò L q ( ; L(S1 ; S3 )) thanks to (3.197b) and Proposition 3.43(iii).
3
3
1
Ë
232
3 Young Measures and Their Generalizations
y N
'
(
'
N (y % " y ; ) " N ' (y; ) " Ù0 " ' (y % " y ) " ' y Ǳ # # lim " Ù0 "
y; ) ( y ) # lim
((
' r y)  y ) Ǳ ;
whi h is just the orresponding omponent in (3.198) after a trivial rearrangement, proving that y N
'
is the Gâteaux dierential.
By (3.197 ), one an also estimate
" ' " " y N (y1 ; ) " " "
#
sup
y L q ;S1
¢
(
)
¢1
" " y N ' (y ; )""""
"L( L q ( ;S1 ); L ( ;S3 )) " ' r y2 ) Ǳ )  y """"L ( ;S ) 3
2
" " " (( ' r " "
y " 1
" ( q "2 )/ " " % b4 y2 S(q1"2)/ % 4 h p(q"2)/q Ǳ " " a 4 % b 4 y 1 S1 y L q ;S1 ¢1 " " "  y 1 " y 2 S y S " 1 1" " "L ( )
sup (
)
¢
" " ( q "2 )/ " " % b4 y2 (Sq1"2)/ % 4 h p(q"2)/q Ǳ """""L q q"2 y1 " y2 L q ( ;S1 ) " " a 4 % b 4 y 1 S1 " ( ) /(
)
;
' ; ) already follows. In parti 
from whi h the lo al equiLips hitz ontinuity of y N (ular, we showed that y N
'
is even the Fré het dierential.
By (3.197b) one an strengthen (at least if
" "
1
"
0
¡ 1) the hypothesis (3.192 ) as follows " "
'(x ; r1 ; s) " '(x ; r2 ; s) S3 # """"X ' r (x ; r1 % a(r2 " r1 ); s)(r2 " r1 ) da """"
¢ a (x) % 3
q " )/ b3 r1 S1 % (
q " )/ b3 r2 S1 %
From this we an estimate, for any
(
" S3 p ( q " )/ q
3 s S2 r 1 " r 2 S1 :
y1 ; y2 ò L q ( ; S1 ), 1 ; 2 ò H
*
(3.200)
p
ò Y H ( ; S2 ),
, and
" " ' ' " " " " # """(' y1 " ' y2 )Ǳ """"L ( ;S3 ) " " "[ N ( y 1 ; 1 )℄( ) " [ N ( y 2 ; 2 )℄( )" "L ( ;S3 ) " " " " ¢ """"(a3 % b3 y1 S(q1")/ % b3 y2 S(q1")/ % 3 h p(q")/q Ǳ ) y1 " y2 S1 """" " "L ( ) " " ( q " )/ ( q " )/ " " " " % b3 y2 S1 % 3 h p(q")/q Ǳ """L q q" y1 "y2 L q ( ;S3 ) ¢ """a3 % b3 y1 S1 ( )
/(
whi h shows the mapping (
)
;
y; ) ÜÙ [ N ' (y; )℄( ) to be lo ally Lips hitzian.
Furthermore, let us estimate
" ' " " y N (y1 ; 1 ) " " "
¢ % The term
I1
" y N ' (y ;
" " " " " "L( L q ( ;S
2)
1 ); L ( ;S3 )) " " ' " " ' " y N ( y 1 ; 1 ) " y N ( y 2 ; 1 )" " " " " " "L( L q ( ;S1 ); L ( ;S3 )) " " ' " " ' " " " "L( L q ( ;S ); L ( ;S )) " y N ( y 2 ; 1 ) " y N ( y 2 ; 2 )" " " 2
1
3
#: I % I :
was already estimated above by means of (3.197 ), while
estimated by means of (3.199b) as follows
1
I2
2
an be now
Ë 233
3.6 Extensions of Nemytski mappings
I2 #
" " " "( ' r
sup
y L q ;S1
¢
(
)
¢1 "
sup
´ g; ( ' r
*
(
(

y Ǳ ( " 2
1
"
y """"L
2 ))
(
;S3 )
yµ ¢

sup
´ 1 y L q ;S1 ¢1 g L ;S3 ¢1
"
" " " 2 """H """"g ('r y2 ) y """"H *
¢ C( y
" ; g  (' r y
2
2)

yµ
)
(
)
sup
2 ))
" " " " 1 y L q ;S1 ¢1 g L ;S3 ¢1
¢
1
)
(
2
y L q ;S1 ¢1 g L ;S3 ¢1
y Ǳ ( "
*
(
2
)
" " L q ( ;S1 ) ) " " 1
"
2
" " " "H
*
;
)
*
(
)
'
proving thus the lo al Lips hitz ontinuity of y N .
'
Eventually, let us prove the ontinuous dependen e of N . For
L q ( ; S1 ) and 1 ; 2 ò H
*
" ' " " " " N ( y 1 ; 1 ) "
" N ' (y ;
#
y " ' y ) Ǳ ; g #
sup (' H ¢1 g L ;S3 ¢1
¢
2
1
" " " " " "L( H ; L ( ;S3 ))
2)
2
*
(
sup
H
¢1
" " " "( ' y 1
 (
" 'y ) Ǳ """"L S3 2
(
;
)
' y1 " ' y2 )
*
*
(
)
H g  (' y1 " 'y2 ) H ¢ C y1 L q
*
sup ; g H ¢1 g L ;S3 ¢1
)
H ¢1 g L ;S3 ¢1
# sup
*
*
y1 ; y2 ò
, we an estimate by means of (3.199a):
*
(
;S1 ) % y 2 L q ( ;S1 ) y 1 " y 2 L q ( ;S1 ) ;
*
*
(
)
whi h shows that N
'
Å
is lo ally Lips hitz ontinuous.
Remark 3.104 (The ase q # %).
The pre eding two lemmas hold also for
q # %
but the assumptions (3.192b), (3.192 ), (3.197b), and (3.197 ) must be suitably modied;
b r
b r
namely the resulted terms of the type
R ÙR %
%
b:
S1 must be repla ed by ( S1 ) with arbitrary in reasing ontinuous fun tion. Also the proofs must be suitably
# 1, the norms of L q ( ) and L q q"2 ( ) must be ' repla ed by the norm in r a( ) and, likewise, y N may be valued in G , supposing additionally that H is G invariant.
modied; for example, if also
/(
)
*
Notation 3.105 (A shorthand onvention: CAR lasses). For the notational simpli ity, H , the mapping ' : , S , S Ù S belongs to the lass
we will say that, for a given
1
CARqH p ( ,S ,S ; S ;
;
1
if
'
2
2
3
3)
is a Carathéodory mapping satisfying the quali ation hypothesis (3.192). If
additionally
' satises also (3.196), (3.197), and (3.199), then we say that ' belongs to
the lass
CARqH pdi ( ,S ,S ; S ) : ;
;
If
S1 # S11 , S12
and, instead of
;
1
2
3
(3.201)
L q ( ; S11 , S12 ) Ê L q ( ; S11 ) , L q ( ; S12 ), , L q2 ( ; S12 ), we will work with
q we need an anisotropi spa e L 1 ( ; S11 )
Ë
234
CARqH1 q2 ;
;;
3 Young Measures and Their Generalizations
p;
(
,S11 ,S12 ,S2 ; S3 )
q2 CARqH1 di ;
or
;;
p;
;
(
,S11 ,S12 ,S2 ; S3 )
for whi h the
onditions (3.192) or (3.196), (3.197), and (3.199) are modied straightforwardly; details are omitted. On the other hand, if
' # '(x ; r) does not depend on svariable, H
q; p; q; p; be omes irrelevant and, instead of CAR ( ,S1 ,S2 ; S3 ) or CAR H H ; di ( ,S1 ,S2 ; S3 ), q ; q;
CAR ( ,S ; S ) or CARdi ( ,S ; S ), respe tively. Let us note that, CARp ( ,S; R) is just Carp ( ; S) dened in (3.81).
we will write just in parti ular,
1
3
1
3
;1
The last onditions, namely (3.199), involve expli itly the norm of
H . The following
example demonstrates that, in fa t, neither (3.199a) nor (3.199b) represent any further restri tion on
' if one hooses a su iently oarse norm on H , as always possible.
Example 3.106 (A universal approa h). We shown in Example 3.76 that every p spa e H of Car ( ; S ) an ertainly be normed by the universal (semi)norm
sub
2
h H #
inf
:(x ; s)ò ,S2: h(x ; s)¢a(x)%b s Sp 2
a L1
(
%b:
)
(3.202)
H #Carp ( ; S2 ), whi h obviously re
In parti ular, this hoi e enables to norm also
 ompa ti ation from the investigated lass. show that (3.199a) will be fullled whenever ' satises
ates the nest onvex We want to
(3.196) and
(3.197b). Indeed, by (3.197b) one gets the estimate (3.200), from whi h we an further estimate
!! ! !! g ( ' y 1 " ' y 2 )!!!
¢ g S3 a % b y 3
*
¢ C y " y 0
1
L q ( ;S1 ) g S
*
3
3
¢ C y " 1
0
q " )/ S1
q " )/ S1 (
2
% b y
(
1
2
% b y
3
3
% (a % b y 3
3
% s Sp 1q" (
)/
3
y2 L q ( ;S1 ) g S 3
*
%
q " )/ S1 (
2
q " )/ S1
q q /( q " ) )
q
y " y2 S1 q
y "y 2 L q
(
and
C0
depending on
q
y 1 " y 2 S1
q
y " y2 S1
1
%
% a q
;S1 )
3
C0
)/
q
y "y2 L q
1
/(
(
;S1 )
q")
3
% b y
for suitable onstants
(
3
(
1
1 1
% s Sp 2q"
1
q S1
% b y 3
2
q S1
% s Sp 2 3
p, q, and only. Taking into a
ount
our hoi e (3.202), we get immediately the estimate
g  (' y1 " ' y2 ) H ¢ C0 y1 " y2 L q
" " " "
" "
¢ C y " y
0
1
g S %
*
3
q y1 " y2 S1 q y 1 "y 2 q L ( ;S1 )
L q ( ;S1 ) g L ( ;S ) 3
2
(
;S1 )
% a q
/(
q")
3
% 1 % a
3
% b y 3
1
q S1
% b y 3
2
q " " " " S1 " "L 1 ( I )
% 3
q /( q " ) L q q" ( ) /(
%b y 3
)
1
q L q ( ;S1 )
% b y 3
2
q L q ( ;S1 )
%
3
;
from whi h we obtain already the assumption (3.199a) with
q /( q " ) q" ( )
C(r) # C0 2 % a3 L q
/(
)
% b rq % 3
3
:
(3.203)
Ë 235
3.6 Extensions of Nemytski mappings
Furthermore, (3.197b) together with (3.197a) also guarantees the assumption (3.199b). Indeed, likewise previously one an estimate
!! !! g
 (
' r y)  y !!!! ¢ g S
*
3
¢ C ¢ C
0
(
)/
g S
% y Sq 1 % a q
(
3
*
3
q
)/
/(
q")
3
y S1
p ( q " )/ q q /( q " )
3 s S
3
3
)/
3
% a % b y Sq1" %
*
(
3
g S
0
% b y Sq1" % s Sp 2q"
a3
% y Sq 1
2
% b y Sq 1 % s Sp 2 : 3
3
Then one gets
"
*
"
¢C
3
"
q /( q " )
q
g  (' r y)  y H ¢ C0 """" g S % y S1 % a3
g % y qL q ( ;S1 ) % L ( ;S3 )
% b y Sq 1 """""L1 3
0
(
)
q /( q " ) q" ( ) %
a3 L q
/(
)
%
3
q
b3 y L q
(
;S1 ) %
3 ;
and therefore
sup
y L q ;S1 ¢1 g L ;S3 ¢1
q /( q " ) q" ( )
g  (' r y)  y H ¢ C0 2 % a3 L q
/(
)
% b y qL q 3
(
;S1 )
%
;
3
)
(
*
(
)
whi h veries already the assumption (3.199b) with
C given again by (3.203).
Analogously, one an also show the estimate
sup
y L q ;S1 ¢1 g L ;S3 ¢1
(
(
g  (' r y1 " ' r y2 )  y H
)
*
¢ C y
)
1
% y
L q ( ;S1 )
2
L q ( ;S1 ) y 1 " y 2 L q ( ;S1 )
;
(3.204)
whi h will be found useful later; f. Example 4.56. Indeed, by (3.197 ) one gets
g  (' r y1 " ' r y2 )  y
¢ g S3a % b y 4
*
¢ C y "y 1
1
4
L q ( ;S1 ) g S3
2
% b y 4
¢
q "2 )/ S1
(
1
*
q "2 )/ S1
(
2
for suitable onstants
q "2 )/ S1 (
2
% (a % b y 4
4
2
(
4
*
%a
4
% s Sp 1q"
C1 y1 "y2 L q ( ;S1 ) g S 3
% b y
%
q y S1
q /( q "2 ) 4
%
%
% s Sp 2q" (
2
)/ q
3
y 1 " y 2 S
1
q "2 )/ S1
y S1
(
1
)/ q q /( q "2 ) )
%
q
y "y2 S1
1
q
y "y2 L q
1
% y Sq 1
(
;S1 )
q
y "y 2 L q
1
q b4 y1 S1
q
y "y2 S1
1
%
;S1 ) q b4 y2 S1 (
% s Sp 2 4
C1 and C1 depending on p, q, and only. This results to
q
;S1 ) g L ( ;S3 ) % y L q ( ;S1 ) % 1 /( q "2 ) % a4 q % b4 y1 qL q ( ;S1 ) % b4 y2 qL q ( ;S1 ) L q q"2 ( )
g  (' y1 " ' y2 ) H ¢ C1 y1 "y2 L q
/(
(
)
%
4
;
Ë
236
3 Young Measures and Their Generalizations
whi h yields (3.204) with
q /( q "2 ) q"2 ( )
C(r) # C1 3 % a4 L q
/(
)
% b rq % 4
4
:
q # %, all these estimates go through with the hypotheses (3.197b, ) modied
If
in the spirit of Remark 3.104.
Remark 3.107.
p
One an observe that the spa e Car (
First, its norm (3.202) ensures the mappings
y :
L q ( ; S
1)
Ù H
y ÜÙ g
; S2 ) is very natural, indeed. ' y) and y Ù Ü g  (' r y)
 (
to be (strong,strong) ontinuous; f. (3.199a) and (3.204), re
spe tively. Se ondly, the assumptions (3.192a), (3.196), and (3.197a) are void provided
H #Carp ( ; S2 ). Indeed, for any y ò L q ( ; S1 ) and g ò L ( ; S3 ), the growth ondition p p (3.192b) ensures ' y S3 òCar ( ; S2 ) so that ertainly g ( ' y ) òCar ( ; S2 ), and thereq fore both (3.192a) and (3.196) are satised. Likewise, for any y ò L ( ; S1 ), the growth q q " p
ondition (3.197b) ensures ' r y L S1 S3 ò Car ( ; S2 ) so that, for any g ò L ( ; S3 ) q p and y ò L ( ; S1 ), we have g  ( ' r y )  y òCar ( ; S2 ) and (3.197a) is satised, as well.
/(
)
;
(
)
Remark 3.108 (Counterexamples for smoothness).
Smoothness
of
Nemytski map
pings is, in fa t, quite strong property and the relaxation whi h makes them partly ane (hen e smooth) is thus worthy also from this analyti al reason. For example,
f ò C (R; R) with the growth at most linear but not ane, the superposition p p operator N f : u ÜÙ f u : L ( ) Ù L ( ) is not ontinuously dierentiable. Indeed,
for any
one an see that
" " " " " "L( L p ( ); L p ( )) "Nf ( u )"Nf ( u )"
(
whi h an be pushed to 0 for
3.6.
1
p " ! !p " " X !!!( f ( u )" f ( u )) w !!! d x # " " " f ( u )" f ( u )" "L
w p L ¢1
# sup
(
) ;
)
u Ù u only if f # onstant, hen e f
is ane.
Twoargument mappings: biane extensions
Let us end this se tion with a biane extension of the Nemytski mapping
L q ( ; S1 ) , L p ( ; S2 ) Ù L ( ; S3 ) with ' : , S1 , S2 Ù S3
N' :
a Carathéodory map
ping satisfying again the growth ondition (3.192b). For this reason, let us have two
C( )invariant subspa es H1 Carq ( ; S1 ) and H2 Carp ( ; S2 ) and assume
:s ò S :s ò S :g ò C( ; S ) : g  '(; ; s ) ò H and g  '(; s ; ) ò H ; *
1
1
2
2
2
1
3
1
2
(3.205)
s1 and s2 in pla e of r and s '(; ; s2 ) Ǳ 1 ò L ( ; S3 ) and '(; s1 ; ) Ǳ 2 ò q p L ( ; S3 ) for any 1 ò YH1 ( ; S3 ) and 2 ò YH2 ( ; S3 ) provided ¡ 1. For # 1 the
in a
ord with the notation from Chapter 7, we will use in the rest of this se tion. Then one has
Ë 237
3.6 Extensions of Nemytski mappings
same holds true with
r a( ; S
in pla e of
3)
L ( ; S3 ). The ondition (3.205) is trivially
satised105 in the following situation:
Lemma 3.109. If S is reexive, ' satises (3.192b) and has the form '(x ; s ; s ) # h (x ; s ) % h (x ; s ) with h ò H and h ò H , then the Nemytski mapping N ' admits a ' q p biane jointly ontinuous extension N from YH ( ; S ) , YH ( ; S ) to L ( ; S ) (or 1 2 to r a( ; S ) if # 1) given obviously by the formula 3
1
1
2
1
2
1
1
2
1
2
2
2
3
3
'
N (1 ; 2 ) # h1 Ǳ 1 % h2 Ǳ 2 :
(3.206)
To extend nonadditively oupled Nemytski mappings, we must onne ourselves to
¡ 1; f. also Example 3.115 below. Putting ' 1 (x ; s2 ) # ['(; ; s2 ) Ǳ 1 ℄(x)
and
' 2 (x ; s1 ) # ['(; s1 ; ) Ǳ 2 ℄(x) ;
(3.207)
we further suppose
: ò YHq1 ( ; S ) : ò YHp2 ( ; S ) :g ò C( ; S ) : g  ' 1 ò H and g  ' 2 ò H : *
1
1
2
2
2
3
(3.208)
1
Lemma 3.110 (Biane extensions of Nemytski mappings). Let S and S be nitedimensional, S be reexive, H and H be separable, ¡ 1, and ' satisfy (3.192b), 1
3
1
2
2
(3.205), and (3.208). Then the following ommutativity property holds
' 1 Ǳ 2 # ' 2 Ǳ 1 and the Nemytski mapping
ontinuous extension
'
N :
N'
in
L ( ; S3 )
(3.209)
,
has a biane separately (weak* weak*,weak)
p q YH1 ( ; S1 ) , YH2 ( ; S2 )
Ù L ( ; S
3)
dened by the formula
'
N (1 ; 2 ) # ' 1 Ǳ 2 :
(3.210)
¡ 1, the growth ondition (3.192b) ensures ' to have a lesser growth q and p in the variables s1 and s2 , respe tively. Then we an repla e 1 and 2 by their q  and p non on entrating modi ations 1 and 2 whi h do exist by Proposition 3.81 be ause H 1 and H 2 are supposed separable and S1 and S2 nitedimensional. 1 By Proposition 3.78, 1 and 2 admit Youngmeasure representations ò Y q ( ; S1 ) 2 p ò Y ( ; S2 ), respe tively. and From (3.207) we obtain by Proposition 3.78 for a.a. x ò :
Proof. Sin e than
' 1 (x ; s2 ) # X '(x ; s1 ; s2 ) S1
105
x (d s 1 )
1
Stri tly speaking, this is true if both
however.
and
' 2 (x ; s1 ) # X '(x ; s1 ; s2 ) S2
x (d s 2 ) :
2
H1 and H2 ontain L1 ( ) 1, whi h an be always supposed,
Ë
238
3 Young Measures and Their Generalizations
It is obvious that
' 1
and
' 2
' 1 Ǳ 2 (x) # ' 2 Ǳ 1 (x) #
p/ p and q/ q, respe tively. Then
has the growth
(3.208) yields the formulae for a.a.
x ò :
x (d s 1 )
X X S2 S1
'(x ; s1 ; s2 )
1
X X S1 S2
'(x ; s1 ; s2 )
2
x (d s 2 )
2
x (d s 2 )
and
(3.211a)
x (d s 1 ) :
1
(3.211b)
By (3.192b) we an estimate
X X ' ( x ; s 1 ; s 2 ) S x (d s 2 ) 3 S1 S2
x (d s 1 )
2
¢
X X a 1 ( x ) S1 S2
# a (x) % b 1
for a.a.
1
1
% b s 1
X s1 S1
1
q/
x
,
2
1
x (d s 1 )
q/
1
%
1
2
p/
x (d s 2 )
2
X s2 S2
x (d s 1 )
1
x (d s 2 )
p/
2
%
x ò . Then we are authorized to use the Fubini theorem, whi h ensures the '(x ; s1 ; s2 )( 1x , 2x )(ds1 ds2 ), with S ,S
both righthand sides in (3.211) to be equal to P 1
% s
1
2
x denoting the standard produ t of the measures
1
2
x and
x . Thus (3.209) has
been proved. Then by (3.208) and by Proposition 3.43 (generalized for the
1 ÜÙ
' 2
Ǳ 1 :
q YH1 ( ; S1 )
Ù
L ( ; S
3)
and
2 ÜÙ
' 1
Ǳ 2 :
S3 valued ase) both Ù L ( ; S3 )
p YH2 ( ; S2 )
are ane and (weak*,weak) ontinuous.
1 # iH1 (u1 ) with u1 ò L q ( ; S1 ) and 2 # i H2 (u2 ) with u2 ò L p ( ; S2 ), one has for any g ò C ( ; S3 ): Moreover, for
*
¼ g; N
'
(
1 ; 2 )½ #
g; ' 1
Ǳ 2 # 2 ; g  ' 1
# ´ i H2 (u ); g  ' 1 µ # 2
# whi h shows that sion of
X g ( x ) ' 1 ( x ; u 2 ( x )) d x
X g ( x ) ' ( x ; u 1 ( x ) ; u 2 ( x )) d x
# g; N' (u ; u 1
'
N (i H1 (u1 ); i H2 (u2 )) # N ' (u1 ; u2 ), so that N
'
2)
;
is a tually an exten
Å
N ' , as laimed.
q # % or p # %, the respe tive terms of the type  % should % % be repla ed by b (  ) with an arbitrary nonde reasing ontinuous b : R Ù R ; f.
Remark 3.111.
If
also Remark 3.104.
Notation 3.112.
In ase
N'
possesses a biane extension, let us agree to write
'
N (1 ; 2 ) # ' Ǳ 1 Ǳ 2 :
(3.212)
This notation is to indi ate that the mapping (
'
'; 1 ; 2 ) ÜÙ N (1 ; 2 ) is, in fa t, tri , S1 , S2 Ù S3
linear. Also let us abbreviate the lass of Carathéodory mappings satisfying (3.192b), (3.205), and (3.208) by
CAR qH1p H2 ( , S , S ; S ) : ;
;
;
1
2
3
(3.213)
3.6 Extensions of Nemytski mappings
Remark 3.113.
Lemmas 3.109 and 3.110 an be ombined together. Thus we an get a
biane separately ontinuous extension of
N'
for any
'(x ; s1 ; s2 ) # '0 (x ; s1 ; s2 ) % h1 (x ; s1 ) % h2 (x ; s2 ) q; p; '0 òCAR H1 ; H2 (
Although
'
Ë 239
, S , S ; S ); h ò H ; h ò H : 1
2
3
1
1
2
(3.214)
2
CAR qH1p H2 ( , S , S ; S
1 2 3 ) for ¡ 1 be ause it may q and p in the variables s1 and s2 , respe tively, the biane
need not belong to
have the growth pre isely
with
;
;
;
separately ontinuous extension does exist, and is obviously given by the formula
' Ǳ 1 Ǳ 2 # '0 Ǳ 1 Ǳ 2 % h1 Ǳ 1 % h2 Ǳ 2 :
(3.215)
' y instead of ' as we did in Se tion 3.6.b. Thus we obtain semibiane extension ' y Ǳ 1 Ǳ 2 . Moreover, we an also use it for
Example 3.114 (Failure of the joint ontinuity).106 An intera tion of os illations generally prevents the extended Nemytski mapping to be jointly ontinuous. Let us demon
# (0; 1), S1 # S2 # S3 # R, q"1 % p"1 ¢ "1 , H1 q , H 2 ontaining L (0 ; 1) R , and ' given by
strate it on a simple example, using
p
ontaining L (0 ; 1)
R
*
*
'(x ; s1 ; s2 ) # s1 s2 :
(3.216)
Let us note that (3.192b) is satised by the Hölder inequality, (3.205) holds trivially,
' 1 (x ; s2 ) # ['(; ; s2 ) Ǳ 1 ℄(x) # [(1 id) Ǳ 1 ℄(x)s2 # g(x)s2 for some g ò L q (0; 1) so that obviously ' 1 ò H2 and analogous onsiderations qp yield also ' 2 ò H 1 . Altogether, we an see that ' òCAR H 1 H 2 ((0 ; 1) , R , R; R). Let us k q now take u 1 ò L (0 ; 1) dened by
and also (3.208) is valid be ause
;
;
;
u1k (x) #
1 "1
if
; x ò (0; 1/2) ; l ò N : x # 2 xl/k ;
otherwise
(3.217)
;
see the lefthand part of Figure 3.15. Furthermore, let us dene a shifted fun tion
L q (0; 1) by
u 1 (x) # u1 (x % 1/k) and u2 k
k
ò L p (0; 1) simply by
k
u 1k ò
u1 # u2 . It is left as an easy k
k
exer ise to verify that
i H1 (u1k ) #
w*lim
i H2 (u2k )
k Ù
k Ù
If
N'
1 1 i (1) % i H1 ("1) #: # 2 H1 2 1 1 # i H2 (1) % i H2 ("1) #: : 2 2
w*lim
1
'
'
lim N (i H1 (u k ); i H2 (u k )) # N ( ;
106
k Ù
i H1 ( u 1k ) ;
(3.218a)
(3.218b)
2
would have the jointly ontinuous extension
k Ù
w*lim
1
2
1
2)
N ' , then inevitably '
# lim N (i H1 ( u k ); i H2 (u k )) ; k Ù
1
2
In onne tion with game theory, this lassi al example an be found basi ally also in Balder
[55, Example 2.6℄, Krasovski and Subbotin [443, Se t. 9.1.2℄, Subbotin and Chentsov [737, Se t. VI.1℄, Warga [791, Se tions IX.2 and X.0.1℄, et .
240
Ë
3 Young Measures and Their Generalizations
N ' (u1k ; u2k ) # u1k u2k # 1 while # u 1 u2 # "1 for any k ò N.
whi h does not hold be ause the lefthand side equals to
k ' k the lefthand side equals to N ( u 1 ; u 2 )
k k
Example 3.115 (Failure of the separate ontinuity).
The
assumption
¡
1
in
Lemma 3.110 is really ne essary be ause otherwise the os illation ee ts in one variable, say
u1 , may intera t with
the on entration ones in the other variable, i.e.
u2 , to prevent even the separate ontinuity. To demonstrate it, let us take # (0; 1), S1 # S2 # S3 # R, and q # p # 2, # 1, and ' given by '(x ; s1 ; s2 ) # PSfrag repla ements Furthermore, let us take the sequen es {
u1k
S
s1 s22 : 1%s1
(3.219)
u1k }kòN and {u2k }kòN a
ording to Figure 3.15. u2k
$k
1/k 1
0
0
1
1/k
"1 Fig. 3.15:
Os illating and on entrating sequen es that intera t via Nemytski mapping.
More pre isely,
u1k ò L2 (0; 1)
is dened again by (3.217) while
u2k ò L2 (0; 1)
is
dened now by
u2k (x) #
$k
if
0
0 x 1/k ;
otherwise
(3.220)
:
!! s
!
s22 /(1%s1 )!!!! ¢ s22 , ' satises (3.192b) with # 1, a 1 # b 1 # 0, and 1 # 1. Taking H # H 1 # H 2 # C ([0 ; 1℄) 0 2 Ô (R ) with R being the smallest omplete subring of C (R), we an identify ea h 2 ò Y H (0; 1; R) with a DiPernaMajda measure ( ; ) ò DM2R (0; 1; R); f. Se tion 3.2. Then 1 from (3.218a) has the representation ( ; ) with Let us note that, thanks to the obvious estimate !! !
# 2;
x
1
1 1 # Æ % Æ" 2 2 1
1
2 # w*limkÙ i H (u2k ) (note that this limit does exist be ause every v ò R has a limit at innity) has the representation ( ; ) with
f. (3.56), while
# 1 % Æ0 ;
x
#
Æ0 Æ
f.
(3.57).
Let
us
note
s1 (1%s1 )"1 Æ0 ò Ö L1 (0; 1)
that
(3.205)
is
if if
x #Ö 0 ; x # 0;
trivially satised,
but
'(; s1 ; ) Ǳ 2
#
so that (3.207) looses a sense and thus the existen e of a
separately ontinuous extension
N
'
is not guaranteed by Lemma 3.110.
3.6 Extensions of Nemytski mappings
Ë 241
In fa t, su h extension does not exist, otherwise it would have to hold 1
'
1
lim lim X '(x ; u k ; u l ) dx # X N ( ; ) dx # lim lim
k Ù l Ù
1
1
0
2
2
1
l Ù k Ù
0
X 0
'(x ; u1k ; u2l ) dx ;
0; 1℄. However,
where the entral integral is understood in the sense of measures on [ this is not true be ause the lefthand side an be evaluated as follows 1/
lim
¬ lim X k Ù l Ù 0
u1k (x)l
l
1%u (x) k
dx # lim
k Ù
1
1 1 # ; 2 2
while for the righthand side one has
lim ¬ lim
l Ù
k Ù
1/
l
X 0
u1k (x)l
1%u k (x)
dx # lim 0 # 0 : l Ù
1
Remark 3.116 (Relations with Se tion 2.5). Let us onsider U # L q ( ; S ), U # L p ( ; S ), Y # L ( ; S ), # N ' , and Fl # FH l # l (H l ) % { onstants on U l } with l : H l Ù CBl (U l ) dened by [ l h l ℄(u l ) # P h l (x ; u l (x)) dx, l # 1; 2. Then we an 1
2
1
2
3
relate the extension stated in Lemma 3.110 with the abstra t situation stated in Proposition 2.36: indeed,
1 # (N ' ) 1
y òY *
where
*
from (2.35) take here the form
N ' )1 y* ℄(u2 ) #
X g(x)

' 1 (x ; u2 (x)) dx ;
(3.221a)
N ' )2 y* ℄(u1 ) #
X g(x)

' 2 (x ; u1 (x)) dx ;
(3.221b)
[(
[(
2 # (N ' ) 2
and
is now denoted by
g ò L ( ; S3 )
*
and
1 # 1 1 , 2 # 2 2 . *
*
For
example, (3.221a) follows from the hain:
N ' ) 1 y * ( u 2 ) #
(; u ) # ; (g  (' u )) # # ; g  (' u ) # g; (' u ) Ǳ # g; ' 1 u
(
1 ; y 1
*
2
1
1
2
2
2
1
2
:
2 # i2 (u2 ) implies g  (' u2 ) ò H1 , so that 1 (g  (' u2 )) ò 2 (g  (' u1 )) ò FH2 , whi h veries (2.33). Moreover, 1 (3.208) also implies g  ' ò H2 , hen e 1 y ò FH2 . The linearity of the mapping 1 g # y ÜÙ y is obvious, while its ontinuity as a mapping L ( ; S3 ) # Y Ù FH2 Let us note that (3.208) for
FH1 .
Similarly, we have also
*
*
with
*
FH2
*
*
endowed with the natural seminorms follows immediately from (3.192b) by
the estimate
!! !!X [ g !! !
 (
!!
' 1 u2 )℄(x) dx!!!! ¢ g L !
(
" " " a1
;S3 ) " " " *
% b % q % u /
1
1
p/ " " " " S2 " "L ( ;S
2
q
3)
1 ò YH1 % ( ; S1 ), with a1 , b1 , and 1 oming from (3.192b). Altogether we shown 1 ò L(Y ; FH2 ). Likewise, 2 ò L(Y ; FH1 ), hen e (2.36) has been veried. Eventu
for
;
*
*
ally, the ommutativity (2.37) follows from (3.209) proved in Lemma 3.110.
242
Ë
3 Young Measures and Their Generalizations
Remark 3.117.
From the proof of Lemma 3.110 one an see that the separately ontin
uous biane extension for
N ' : YH1 ( ; S1 ) , YH2 ( ; S2 ) Ù L ( ; S3 ) p
q
# 1 but one must restri t N
'
on
does exist even
q and pnon on entrating generalized Young 2 was not
fun tionals only. This also ex ludes the situation in Example 3.115 where
pnon on entrating.
4 Relaxation in Optimization Theory ...
I
observed
that
the
maximum
prin iple
in
ontrol theory is equivalent to the onditions of EulerLagrange
and
Weierstrass
in
the
lassi al
theory. [384, p. viii℄
Magnus Rudolph Hestenes
(19061991)
I had a proof of the Maximum Prin iple. Not as a su ient ondition, but as a ne essary ondition... [126℄
Vladimir Grigorevi h Boltyansky In
addition
onsider
to
`original'
"approximate"
solutions solutions
(19252019)
..., that
we are
also se
quen es ... and `relaxed' solutions that are a form of weak, or extended, solutions. ... we study relaxed solutions for a number of reasons: they yield a
omplete theory that en ompasses both existen e theorems and ne essary onditions; they provide the means for onstru ting optimal approximate solutions;
they
properly
model
ertain
physi al
situations [791, pp. xixii℄
Ja k Warga (19222011)
This hapter begins with a relaxation theory for abstra t optimization (esp. optimal ontrol) problems: Se tion 4.1 studies basi relations between the original and the relaxed problems and also a omparison of various relaxations is performed there. The exposition in this se tion pro eeds on the most general level, using extensions of the original problems by the onvex ompa ti ation theory from Chapter 2. This enables us also to formulate the rstorder optimality onditions, whi h takes the form of abstra t maximum prin iples if one works with onvex ompa ti ations in their
anoni al forms. The remaining part deals with more on rete optimal ontrol problems with ontrols ranging some Lebesgue spa e. Therefore, the relaxation is performed in terms of the generalized Young fun tionals developed in Chapter 3. Two general important prin iples are treated separately in Se tion 4.2: the rst one on erns Pontryagintype maximum prin iples reated by a lo alization of the integral maximum prin iples resulted straightforwardly from the abstra t maximum prin iples derived in Se tion 4.1, and the se ond one establishes a ertain non on entration regularity of generalized Young fun tionals whi h satisfy these integral maximum prin iples. This se tion further dis usses various onsequen es of the maximum prin iple. All this enables us to treat a wide lass of on rete problems with ontrols ranging Lebesgue spa es by a routine way, assembling already prefabri ated tools and results. Usage of these prefabri ated tools is demonstrated on quite on rete optimal ontrol problems in Se tions 4.36. They on ern su
essively the nonlinear dynami al https://doi.org/10.1515/9783110590852004
Ë
244
4 Relaxation in Optimization Theory
systems (i.e. initial value problems for ordinary dierential or dierentialalgebrai equations), partial dierential equations of the ellipti and the paraboli types, and the Hammerstein integral equation. The rst ase is handled quite omprehensively to demonstrate all possible appli ations of the abstra t results, i.e. impa ts of results for the relaxed problem to the original problem itself, relations between the original and the relaxed problems, existen e of solutions to the relaxed problems and their stability, rstorder optimality onditions, qualitative properties of optimal relaxed ontrols and their numeri al approximation. The resting ases are exposed more briey rather to show various pe uliarities onne ted with the parti ular distributed parameter systems; nevertheless, always the wellposed relaxed problem is onstru ted and a Pontryagintype maximum prin iple is derived. The on rete form of su h maximum prin iples is losely related with the hosen onvex ompa ti ation used to relax the original problem, the Hamiltonians involved in these prin iples having a very denite meaning, namely Gâteaux dierentials in anoni al forms. In all ases, we will admit
ontrols that need not be bounded in the
L
norm; in parti ular we will onsider a
polynomial growth (of possibly dierent orders
p and q) of both the ontrols and the
states. Being usual in on rete problems and not essentially onning appli ability, ex ept Se t. 4.5.a we onne ourselves on the more onventional on ept of sequen es rather than general nets (and thus e.g. subsequen es instead of ner nets used in Chapters 2 and 3). To this goal, essentially without loss of generality, we will assume here separability of spa e of test fun tions.
4.1
Abstra t optimization problems
In this se tion we will develop a relaxation theory of the abstra t optimization problem in the form (PO )
Minimize subje t to
(u) for u ò U ; R(u) ¢ 0 ;
:UÙR a ost fun tion, R : U Ù a mapping, and an ordered Bana h spa e with D
where
U
is a set (say a topologi al spa e) endowed with a bornology
B,
by a " ò D.
a losed onvex one with the vertex at the origin. Re all that the ordering of
one
D
is dened as follows: for
; ò
we write
¢
if and only if
D has a nonempty interior int (D), we will write if and only if " òint (D). Besides, we will onsider a lo ally onvex topology on ner than the weak topology (so that D remains losed with respe t to ); f. also Se tion 1.2.d.
Moreover, if
Without any further data quali ation, (PO ) need not have any solution or the set of solutions
Argmin(PO ), even if nonempty, need not be stable with respe t to data
perturbations. The on ept, more natural than the solutions in the above lassi al sense, relies on asymptoti ally admissible and minimizing sequen es, invented basi
u
ally by Levitin and Polyak [489℄. A sequen e { k } k òN will be alled
asymptoti ally
Ë 245
4.1 Abstra t optimization problems
"D in the topology , whi h means that for any neighbourhood N of 0 there is k N ò N su h that R(u k ) ò N " D whenever k £ k N . For example, if is the strong topology, then {u k }kòN is asymptoti ally admissible if lim k Ù inf ¢0 R(u k ) " # 0. If is the weak topology, then {u k }kòN is asymptoti ally admissible if, for any ò , lim k Ù inf ¢0 £ 0 provided h ò H su h that h(x ; s) £ a0 (x) for some a0 ò L1 ( ), p < " ; h > ¡ 0 provided #Ö and h ò H is oer ive in the sense h ( x ; s ) £ a 0 ( x ) % b s 1 with some a 0 ò L ( ) and b ¡ 0.
£ 0. The point (i) is proved. Let us suppose that (ii) does not hold, so that < " ; h > # 0 for some h ò H
from whi h we obtain in the limit
£ 0. Taking into a
ount also our assumption < " ; h > # 0, we obtain bsp with b p
" ; h ¢ " ; h # 0 :
(4.40)
" h , we an see that ¢
Å
Let us now turn our attention to onsequen es whi h the pointwise maximum prin iple may have in on rete situations. Let us mention two typi al examples.
Example 4.25 (Bangbang ontrols).
If the ontrol a ts linearly both in the ontrolled
p
H # L p ( ; S ) so that Y H ( ; S) # H Ê p ¢ %; for S # Rm f. also Example 3.73. Supposing S measurable with losed onvex values, we have U ad # Uad . For a given p p m Hamiltonian h ( x ; s ) # < g ( x ) ; s > with some g ò L ( ; S ) and an optimal ò Y ( ; R ) H
system and in the ost fun tional, then we an take25 *
23
*
L p ( ; S) provided S is reexive and 1
*
Similar kind of results for spe ial lass of minimization problems has been obtained also by Kinder
lehrer and Pedregal [425℄. For optimal ontrol problems see Berlio
hi and Lasry [114℄ where this was only supposed as a hypothesis, however.
"h required in Lemma 4.22(ii). L p ( ; S* ) with a subspa e of Carp ( ; S) via the mapping g
24
Note that (4.36) ensures just the oer ivity of
25
Here we identify naturally
h(x ; s) # .
ÜÙ h with
4.2 Optimization problems on Lebesgue spa es
identied with some
Ë 269
u ò L p ( ; S), the maximum prin iple (4.34) results in
:a.a. x ò :
g ( x ) ; u ( x )
# max g(x); s :
(4.42)
sòS(x)
Let us note that the maximum in (4.42) is a tually attained, for example, just at the point
s # u(x). Furthermore, (4.42) an be rewritten into the form
:a.a. x ò : g(x) ò N S x (u(x)) ;
(4.43)
( )
whi h implies parti ularly that
:a.a. x ò : u(x) òbd(S(x))
or
g(x) # 0 :
(4.44)
The phenomenon that some optimal ontrol tends to follow the boundary of
S(x)
is
alled a bangbang prin iple.
Example 4.26 (Chattering ontrols). If p # , S # Rm , S(x) is ompa t, ahd H # Car ( ; Rm ), then the orresponding onvex  ompa ti ation of L ( ; Rm ) is m equivalent with the set of the Young measures Y ( ; R ). Taking a Hamiltonian p m m # h(x ; s) ò Car ( ; R ) and identifying ò YH ( ; R ) with a Young measure
x } x ò , then (4.35) results in
{
:a.a. x ò :
h(x ; s) x (ds) X S(x)
# max h(x ; s) :
(4.45)
sòS(x)
S(x) is ompa t x ò , h(x ; ) attains its maximum on S ( x ) at a nite number of points, say s l ( x ) ò S ( x ), l # 1 ; :::; k . Then (4.45) Let us note that the maximum in (4.45) is a tually attained be ause
h(x ; )
and
is ontinuous. It is an often ase that, for a.a.
says that, in parti ular, the optimal relaxed ontrol must be ne essarily a onvex
ombination of the Dira measures:
x
k
k H a l (x)Æ u l (x) l #1
(4.46)
) and u l ò L ( ; Rm ) su h that 0 ¢ a l (x) ¢ 1, u l (x) ò S(x), and a l (x) # 1 for a.a. x ò . In other words, the Young measure is omposed from
with some
k l #1
#
al ò L
(
atoms. Relaxed ontrols of this type are alled hattering ontrols.26 Su h ontrols
are espe ially important if they are pie ewise onstant (resulting, for example, as optimal ontrols for approximate problems reated by the approximation from Se t. 3.5.b be ause then they an be readily implemented on omputers. Paraphrasing this denition for the general ase, a generalized Young fun tional
ò YH ( ; Rm ) will be alled hattering if it admits the following representation p
#
26
H
k a i (u ) ; l #1 l H l
with
u l ò L p ( ; Rm )
and
al ò L
(
)
(4.47)
Chattering ontrols are also o
asionally alled, e.g. in [267, 343℄, sliding modes or regimes, or
o
asionally Gamkrelidze's ontrols. Also, sometimes dierent meaning of hattering ontrols an o
ur; f. Zelikin and Borisov [814℄.
Ë
270
4 Relaxation in Optimization Theory
a l (x) £ 0 and kl#1 a l (x) # 1 for a.a. x ò , where the expression kl#1 a l l with a l ò G and l ò H is dened for H being G invariant by27 su h that
*
:h ò H :
¼H
k a l #1 l l
; h½ #
H
k h Ǳ l ; a l : l #1
(4.48)
Let us note that this extended denition has the previous meaning of a linear ombi
l provided a l are onstants on . in the form (4.47) will be also said katomi with k ò N referring to the number k in (4.47). As every i H ( u l ) is p non on entrating, it an be seen that from (4.47) is p non on entrating, as well. Therefore, by Proposition 3.78, every hattering p ò YH ( ; Rm ) admits a Youngmeasure representation ò Yp ( ; Rm ), and obviously nation of
Every
takes the form (4.46). The following assertion treats situations when every optimal solution is hattering:
Proposition 4.27 (Chattering ontrols I). Let ò U ad YHp ( ; Rm ) satisfy the maximum prin iple (4.32), H be separable, U ad #Ö take the form (4.33) with a measurable losedm valued mapping S : Â± R , and h satisfy the des ent ondition (4.36). Then: (i) If, for a.a. x ò , h ( x ; ) attains its maximum on S ( x ) at no more than k ò N points,
katomi . h(x ; ) is stri tly on ave for a.a. x ò and S is onvexvalued, then is 1atomi . In other words, the relaxed ontrol , being of the form i H ( u ) with some u ò U ad , is, is inevitably
(ii) If
in fa t, an original ontrol. Proof. By Theorem 4.24,
is
pnon on entrating, and therefore it admits a YoungRm ) su h that # P PRm h(x; s) x (ds) dx for
ò Yp ( ;
measure representation
h ò H ; see Proposition 3.78. Moreover28 supp( x ) S(x) for a.a. x ò . By Theorem 4.21, satises also the pointwise maximum prin iple (4.35), whi h results here in P h(x ; s) x (ds) # [h Ǳ ℄(x) # h S (x) :# supsòS x h(x ; s) for a.a. x ò . S x any
(
( )
S (x) where S (x) # {s ò S(x); h(x ; s) #
)
h S (x)}. As
S0 (x) is supposed 0 k points, x takes the form (4.46) with some u l (x) ò S(x) and a l (x) ò [0; 1℄ su h that kl#1 a l (x) # 1. By Theorem 1.25 the multivalued mapping S0 :
Â± Rm is measurable, so that by Theorem 1.26 we may suppose u l measurable. Then also a l may be supposed measurable be ause x ÜÙ x is weakly measurable.29
Then supp( x )
0
to onsist from at most
*
27
Of ourse, the lefthandside duality is between
H * and H while the righthandside one is between
# i H (u l ), then always h Ǳ l ò L ( ) so that the righthand side of (4.48) has a good meaning even for a l ò L ( ) \ G , as used in (4.47). # 0 for h(x ; s) # inf sòS x s " s . Yet, w*limkÙ Æ(u k ) # 28 This fa t is obviously equivalent to h Ǳ for u k ò U ad implies limkÙ h u k # h Ǳ # 0 be ause h u k # 0 for any u k ò Uad ; note also that h is a G* and G. Moreover, if l
1
( )
Carathéodory fun tion be ause
29
We may suppose that all
k
S is measurable. k points { u l ( x )} l#1
are mutually dierent otherwise we an divide
on measurable parts with this property for various
k.
Sin e all
ul
are measurable, there is
hl
ò
4.2 Optimization problems on Lebesgue spa es
Ë 271
u l (x) maximizes h(x ; ) over S(x), it holds h(x ; u l (x)) # h S (x). Simultap neously, by the des ent ondition (4.36), it also holds h ( x ; u l ( x )) ¢ a ( x ) " b u l ( x ) . p S " 1 p Altogether, b u l ( x ) ¢ a ( x ) " h ( x ). Thus we have got u l L p ;Rm ¢ b ( a L 1 % S 1 p p m 1 h L ) %, so that ertainly u l ò L ( ; R ), as laimed in (4.47). The point (i) As always
(
/
)
(
)
/
(
)
has been thus proved. The point (ii) then follows immediately be ause a stri tly on ave Hamiltonian
Å
attains its maximum at no more than one point.
If the Hamiltonian need not attain its maximum at a nite number of points, some optimal relaxed ontrols need not be hattering. Nevertheless, the maximum prin iple enables sometimes to establish a bit weaker result, namely that at least one optimal relaxed ontrol is hattering. Even su h weaker result might be of some usage espe ially if our task is to nd (approximately) not all optimal ontrols, but at least one optimal
ontrol, whi h is a usual standpoint, indeed. Su h kind of results is supported by the following general prin iple:
Proposition 4.28 (Chattering ontrols II).
H
Let
be
separable,
Uad
#Ö
take
S measurable and losedvalued, there exist an optimal30 p 0 ò U ad YH ( ; Rm ) satisfying the maximum prin iple (4.32) with h ò H satisfying the k des ent ondition (4.36), and let, for some nite olle tion { h l } l #1 H the following
the form (4.33) with
impli ation holds:
ò U ad ; h Ǳ # h S with h S (x) :# supsòS h l Ǳ # h l Ǳ 0 ; l # 1; ::: ; k Then there exists at least one
x
( )
h(x ; s) ;
/ 7 ? 7 G
âá is optimal :
(4.49)
whi h is optimal and (k%1)atomi .
0 is pnon on entrating, and therefore by Proposition 3.78, ò Yp ( ; Rm ) su h that < 0 ; h > # P P m h ( x ; s ) x (d s ) d x for any h ò H . Besides, we may suppose that, for
R p S a.a. x ò , supp( x ) is ontained in S 0 ( x ) # { s ò S ( x ); s ¢ ( a ( x ) " h ( x ))/ b } where a and b ome from (4.36). 1% k Let us now dene the multivalued measurable mapping C : Â± R by Proof.31 By Theorem 4.24,
it admits a Youngmeasure representation, i.e. there exists some
C(x) # As
h ( x ; s ) ;
h1 (x ; s) ; ::: ; h k (x ; s) ò R1%k ; s ò S0 (x) :
S0 (x) is ompa t for a.a. x ò , C(x) is ompa t as well. As
Rm
L1 ( ; C0 (
)) su h that
h l (x ; u l (x)) # 1 while h l (x ; u j (x)) # 0 for j
%
x òr a1 ( S 0 ( x )), we have
#Ö l, whi h shows that a l # h l Ǳ
must be measurable, too.
30 31
The adje tive optimal an have an entirely formal meaning in this statement. Some ideas of this proof ome from the work by Bonnetier and Con a [136℄. f. also Balakrishnan
[49, Thm. 1.9.1℄ who did not handle measurability, however.
272
Ë
4 Relaxation in Optimization Theory
g(x) #
#
X h ( x ; s ) x (d s ) ; X h 1 ( x ; s ) x (d s ) ; ::: ; X h k ( x ; s ) x (d s ) S0 (x) S0 (x) S0 (x) [ h Ǳ 0 ℄( x ) ;
[
h1 Ǳ 0 ℄(x); ::: ; [h k Ǳ 0 ℄(x)
ò
o(
C(x)) :
Then, by the Carathéodory theorem 1.12, this point an be obtained by a onvex ombi
k%1 points of C(x) be ause C(x) is a subset of a kdimensional ane h S (x)} , R with h S (x) :# supsòS x h(x ; s). In other words, there k %1 k %1 k %1 exists { u i ( x )} i #1 S 0 ( x ) and { a i ( x )} i #1 [0 ; 1℄ su h that i #1 a i ( x ) # 1 and nation of at most
manifold, namely {
(
[ h Ǳ 0 ℄( x ) ;
#
H
[
)
h1 Ǳ 0 ℄(x); ::: ; [h k Ǳ 0 ℄(x)
k %1 a (x) h(x ; u i (x)); i #1 i
h1 (x ; u i (x)); ::: ; h k (x ; u i (x))
:
g and C are measurable, we an suppose the mappings a i and u i measurable a i ò L ( ) and u i ò L p ( ; Rm ) thanks to the denition of S S0 ; re all that always a ; h ò L1 ( ). Therefore also u i ò Uad . k %1 Let us dene by < ; h > # P i #1 a i ( x ) h ( x ; u i ( x )) d x with h ò H , whi h is ( k %1)
atomi by the very denition. We have also ò U ad be ause the Young measure , k %1 p m dened by x # i #1 a i ( x ) Æ u i x , belongs to Y ( ; R ), determines ee tively the fun tional by means of the relation (3.14), and is attainable from U ad sin e ea h u i ò U ad . S Also we have obviously h Ǳ # h Ǳ 0 # h and h l Ǳ # h l Ǳ 0 for any l # 1 ; :::; k . Therefore, by the hypothesis (4.49), is optimal. Å
As both
( f. Theorem 1.27) and thus
( )
Oneatomi hattering optimal ontrols are naturally of a spe ial importan e, as they are optimal for the original problem. This is another noteworthy appli ation of the relaxed problems. The essen e of su h existen e theory for the original problems
an be seen on a prototype problem
Minimize subje t to
X ' ( x ; y ( x ) ; u ( x )) d x
( ost fun tional)
A(y) # f(y; u) on ; ( x ; y ( x ) ; u ( x )) ¢ 0 :a.a. x ò ; u(x) ò S(x) :a.a. x ò ; y ò L q ( ; R ); u ò L p ( ; Rm ) ;
(state equation)
/ 7 7 7 7 7 7 7
(state/ ontrol onstraints) ? 7 7 ( ontrol onstraints)
(4.50)
7 7 7 7 7 G
R
m 2 is assumed ordered by a one D to give where A is an abstra t operator and where p m ), sense to the inequality in (4.50). Moreover, we will assume that, for any u ò L ( ; the state equation
A(y) # f(y; u)
has a unique solution
y
R
so that the stateequation
itself does not represent any onstraint on the ontrol32 and determines a ontroltostate mapping
32
u ÜÙ y whi h is (weak,strong) ontinuous.
This attribute ex ludes the variational problems where
whi h (ex ept trivial ases) brings impli it restri tions on
Ay # f(x ; y(x); u(x)) takes the form y # u u. Similar impli it restri tions may arise in
optimal ontrol of some dierentialalgebrai equations, f. Se tion 4.3.g below.
4.2 Optimization problems on Lebesgue spa es
Ë 273
a ò L ( ), natural growth onditions for the Carathéodory Rm Ù R, f : ,Rn ,Rm Ù Rn and : ,Rn ,Rm Ù R are
Using the notation integrands
' : ,R
n,
'(x ; r; s) ¢ a1 (x) % brq % sp ; f x ; r; s) ¢ a p1 (x) % br
(
(
q/p1
x ; r; s) ¢ a p2 (x) % br
(4.51a)
p/ p 1
% s ; % sp p2
q/p2
and
(4.51b)
/
(4.51 )
p1 ; p2 ¡ 1 and b; ò R so that '(y; u) lives in L1 ( ) while f(y; u) ò L p1 ( ; Rn ) p ( y; u ) ò L 2 ( ; R ). Moreover, we suppose the oer ivity of (4.50) in the sense
for some and
; ¡ 0 :a.a. x ò : (r; s) ò Rn ,Rm : 1
Let us note that
lim u Lp ;Rm Ù
u
(
)
P
'(x ; r; s) £ 1 sp :
'(y u ; u) dx Ù %,
(4.52)
whi h ensures that every se
quen e of ontrols { k } k òN minimizing for (4.50) is inevitably bounded in
L p ( ; Rm ).
Theorem 4.29 (FilippovRoxin prin iple).33 Let there exist A" : L p1 ( ; Rn ) Ù L p1 ( ; Rn ) ontinuous and ompa t, S : Â± Rm be measurable losedvalued, (4.51) 1
(4.52) hold, and the minimization problem (4.50) be feasible. Let furthermore the so alled orientor eld
Q(x ; r) #
Q dened by
' ( x ; r; s )%
R% ; f x; r; s (
0
)
;
(
x ; r; s)% D ò R1%n%m ; s ò S(x)
(4.53)
be onvex and losed. Then (4.50) has a solution. Proof. First, inspired by [561℄, it will be more suitable to reformulate the onvexity and
losedness of
Q as a ondition
:a.a. x ò : r ò Rn : o ',f , with
( x ; r; R ( x ; r ))
Q(x ; r)
R(x ; r) # s ò S(x);
(
x ; r; s) ¢ 0 :
(4.54a) (4.54b)
q1 ; q2 ò Q(x ; r), one has s ; s ò S(x) su h that q1 £ q2 # f(x ; r; s i ), and q3i £ i 3 ( x ; r; s ) for i # 1 ; 2, and then (4.54) guarantees existen e of s ò S(x) su h that 1 1 i i i i #1 2 2 ( ' ( x ; r; s ) ; f ( x ; r; s ) ; ( x ; r; s )) ò Q , whi h eventually results to i #1 2 2 q i ò Q(x ; r). Conversely, the onvexity and losedness of Q implies (4.54) be ause always
o[', f , ℄(x ; r; S(x)) oQ(x ; r). p m Then, we make the relaxation by hoosing the linear spa e H Car ( ; R ) as Indeed, (4.54) implies the onvexity of 1
i
2
Q
from (4.53) be ause, taking
'(x ; r; s i ),
i
;
;
H # span g0 (' y0 ) % g1  (f y1 ) % g2  ( y2 ) òCarp ( ; Rm );
g0 ò C( ); g1 ò C( ; Rn ); g2 ò C( ; Rn ); y0 ; y1 ; y2 ò L q ( ; R ) :
33
(4.55)
This assertion generalizes the FilippovRoxin ondition formulated originally for un onstrained
optimal ontrol of ordinary dierential equation [311, 704℄, f. also Cesari [193℄ or Mordukhovi h [549, 551℄.
Ë
274
4 Relaxation in Optimization Theory
Then we onsider the relaxed problem
/ 7 7 7 7
( ost fun tional)
A(y) # f y Ǳ on ; y Ǳ ¢ 0 on ; y ò L q ( ; Rn ); ò U ad ;
subje t to
where
' y Ǳ dx
X
Minimize
(state equation)
(4.56)
?
7 (state/ ontrol onstraints) 7 7 7 G
U ad Y H ( ; Rm ) is from (4.30) with H from (4.55). p
H from (4.55), we p1 ; p2 ¡ 1 so that, by (4.51) (4.52), f ( x ; r; ) and ( x ; r; ) has a lesser growth than ' ( x ; r; ), any any solution to p (4.56) is p non on entrating and thus has a representation by an L Young measure p m ò Y ( ; R ), f. Proposition 3.78. This solves the following problem: By using Proposition 3.102 modied straightforwardly for spa e
an see that
H
is separable. Furthermore, using that
X X
subje t to
A(y) # X
Rm
Rm
X
(
Rm
x (d s )d x
'(x ; y(x); s)
Minimize
x (d s )
x ; y(x); s)
R
y ò L q ( ; Let us note that, for a.a.
x (d s )
f(x ; y(x); s)
¢0
ò Yp ( ;
n );
R
;
supp x S(x) :a.a. x ò :
x ò , the probability measure
ò Yp ( ;
xò
Let us x have
limÙ
restri tion with
P m R \B
p ; y) ò Y H ( ; m ) solves (4.57).
k x;
#
k x; For
£ 1,
x (d s )
p
)
(
; Rn ). Its
ds) %. By the Lebesgue theorem, we
p
x(
# 0, where B
is the ball in
Rm of the radius . The
x ; ; f. the proof of Theorem 3.6. Then we put
#
k x;
% Æ s0 0
x (d s ) ;
0 # 0 () # X
and
Rm \ B
0 () ¢ PRm
ertainly P m [ ' , f , R
obviously
and therefore
Rm , L q
i # i (x) £
limkÙ
s
R
G
k k x B an be approximated by a k atomi measure x ; # i #1 i Æ s i k 0, i#1 i # PB x (ds), and s i # s i (x) ò B S(x), so that w*
#
x;
s
for whi h P m R
(4.57)
x must be supported on the
By the assumed oer ivity, (4.56) has a solution (
L p Youngmeasure representation
? 7 7 7 7 7 7 7
:a.a. x ò ;
m );
R(x ; y(x)) dened above in (4.54b).
losed set
on
/ 7 7 7 7 7 7 7
\
s0 ò S(x):
k x (d s ). Let us note also that i #0 i # 1 k ℄( x ; y ( x ) ; s ) x ; (d s ) ò o[ ' , f , ℄( x ; y ( x ) ; S ( x )). B
s
p
Moreover,
lim
k Ù
X
Rm
v(s)
k x ; (d s )
# X v(s) Rm
for any
# lim
k X v ( s ) x ; (d s ) k Ù B
x ; (d s )
% v(s
% v(s ) # X v(s) 0
0
Rm
0
x (d s )
0)
%X
v(s)
Rm \ B
v ontinuous. If v has at most pgrowth, we have limÙ
v # [', f ,
℄(
P m R \B
% v(s
v(s)
0
p s
0)
x (d s )
x (d s ) %. Also lim Ù 0 ( ) v ( s 0 ) x ; y(x); ), by (4.54) we obtain
by the Lebesgue theorem be ause P m R Altogether, for
x (d s )
#0 # 0.
4.2 Optimization problems on Lebesgue spa es
X
Rm
',f ,
( x ; y ( x ) ; s ) x (d s )
# lim lim Ù
Rm
ò o ', f ,
k ( x ; y ( x ) ; s ) x ; (d s )
', f ,
X
k Ù
Ë 275
( x ; y ( x ) ; S ( x ))
Q(x ; y(x)) :
(4.58)
Let us put
U(x) # s ò S(x); '(x ; y(x); s) ¢ X
'(x ; y(x); )
S(x)
f(x ; y(x); s) # X
x (d ) ;
f(x ; y(x); )
S(x)
(
x ; y(x); s) ¢ X
S(x)
(
x (d ) ;
x ; y(x); )
x (d ) ;
(4.59)
U(x) is nonempty: Indeed, by (4.54), for any (q0 ; q1 ; q2 ) ò Q(x ; y(x)) s ò S(x) su h that q0 £ '(x ; y(x); s), q1 # f(x ; y(x); s), and q2 £ (x ; y(x); s).
and show that there is
Hen e, for the parti ular hoi e
(
q0 ; q1 ; q2 ) # q0 (x); q1 (x); q2 (x) # X
S(x)
',f ,
( x ; y ( x ) ; s ) x (d s ) ;
(4.60)
q0 (x) £ '(x ; y(x); s), q1 (x) # f(x ; y(x); s), and q2 (x) £ x ; y(x); s) for some s ò S(x), hen e U(x) #Ö . m dened by (4.59) is meaMoreover, the multivalued mapping U : Â± R surable. Indeed, weakly* measurable and ' , f and Carathéodory mappings imply that q from (4.60) is measurable. Furthermore, by [37, Thm. 8.2.9℄, the level sets x ÜÙ {s ò Rm ; '(x ; y(x); s) ¢ q0 (x)}, x ÜÙ {s ò Rm ; f(x ; y(x); s) # q1 (x)}, and x ÜÙ {s ò Rm ; (x ; y(x); s) ¢ q2 (x)} are measurable. By [37, Thm. 8.2.4℄, the interse tion of these level sets, whi h is just U ( x ), is also a measurable multivalued mapping. Obviously, U ( x ) is losed for a.a. x ò . Then, by [37, Thm. 8.1.4℄, the multivalued mapping U possesses a measurable sele tion u ( x ) ò U ( x ). In view of (4.59), f ( y; u ) # q1 # PRm [f y℄(; s) (ds) # f y Ǳ and ( y; u ) ¢ q 2 # P m [ y ℄( ; s ) (d s ) # y Ǳ ¢ 0 so that the pair (u ; y) is adR the in lusion (4.58) implies that (


missible for (4.50), and moreover
X ' ( x ; y ( x ) ; u ( x )) d x
¢ X q (x)dx # X
0
X
Rm
'(x ; y(x); s)
x (d s ) d x
# X [' y Ǳ ℄(dx) # min(4:56) ¢ inf (4:50) :
Eventually, the oer ivity (4.52) with the assumed feasibility of (4.50) implies
1 X u(x)p dx ¢ X '(x ; y(x); u(x)) dx ¢ inf (4:50) %:
p Therefore, u ò L ( ;
R
m ), whi h ompletes the proof that
u solves (4.50).
Employing the maximum prin iple, the FilippovRoxin theory an be rened so that existen e an be obtained even for non onvex orientor elds. Assuming onstraints being qualied, the maximum prin iple for any solution to (4.50) reads as
Ë
276
4 Relaxation in Optimization Theory
:a.a. x ò : h
*
h
;
; y ( x ; u ( x )) *
*
;
; y (x ; s) *
where the adjoint state
[
A
℄
*
*
# max h
*
;
*

(
x ; y(x); s) " '(x ; y(x); s) ;
(4.61a)
solves the adjoint equation
*
for some multiplier
with
*
#  f(x ; y(x); s) "
% [f y y Ǳ ℄ # ' y y Ǳ % [ *
; y ( x ; S ( x ))
*
y Ǳ ℄
*
(4.61b)
£ 0. *
*
Corollary 4.30 (FilippovRoxin prin iple rened).34 Let (4.51)(4.52)
together with the
(here unspe ied) assumptions ensuring the maximum prin iple (4.61) hold. Let also (4.54a) holds for some
R(x ; r) s ò S(x); h
*
;
; y (x ; s) *
# max h
*
;
; y ( x ; S ( x )) : *
(4.62)
Then (4.50) has a solution. Let us note that a very spe ial ase, whi h an be however handled in a simpler way by a dire t method applied to the original problem, appears if
f x ; r; ) and
and (
(
'(x ; r; )
is ane
x ; r; ) are onvex.
Example 4.31 (W.H. S hmidt [699℄, modied). Let us onsider m # n # 1, # (0; 1), A # "d /dx , '(x ; r; s) # "r % s , f(x ; r; s) # sin(s), S(x) # R, # 0. More2
2
3
2
A is now a 2ndorder ellipti operator, we should pres ribe boundary ondiy ò H01 (0; 1) instead of L q ( ; Rn ) in (4.50). Then the orientor eld Q ( x ; r ) is onvex, f. Figure 4.1(left) so Theorem 4.29 an be applied. Note that '(t ; r; ) is not ane so that we annot use simple weak ontinuity arguments of a over, as
tions e.g. by onsidering
dire t method. Let us still modify this example by taking a (non onvex) ontrol onstraint
S(x) # ["3 ; "2℄ [0; ℄. Then the orientor eld Q(x ; r) is no longer onvex,
f. Figure 4.1(right) so that Theorem 4.29 annot be applied. Yet, the adjoint problem
d /dx # "3y , (0) # 0 # (1), hen e always £ 0 everywhere on [0 ; 1℄, so that the Hamiltonian h ( x ; s ) # ( x ) sin( s ) " s annot attain its maximum on ["3 ; "2 ℄ but only on [0 ; ℄. Then the requirement (4.62) is guaranteed and
takes the form
2
*
2
2
*
*
*
*
*
2
therefore Corollary 4.30 an be used.
34
The essen e of involving an information from the maximum prin iple in Corollary 4.30 is to ex
lude values of the ontrol whi h annot o
ur in optimal ontrols anyhow. See also [699℄ for this argumentation in on rete situations.
4.3 Optimal ontrol of nitedimensional dynami al systems
(
repla ements '
'; f)(x ; r; S(x))
Q(x ; r)
(
'; f)(x ; r; S(x))
Ë 277
Q(x ; r)
f
f
non onvex orientor field
onvex orientor field
Fig. 4.1: An example of the graph [( ' ;
f)℄(x ; r; S(x)) and the onvex (left) or non onvex (right) orientor Q(t ; r) guaranteeing existen e of solutions through the FilippovRoxin prin iple, possibly rened for the non onvex Q as in Corollary 4.30.
eld
4.3
Optimal ontrol of nitedimensional dynami al systems
In this se tion we will treat an optimal ontrol problem for a system governed by an initialvalue problem for an ordinary dierential equation (a so alled dynami al system). We want espe ially to demonstrate the omplete analysis of the problem: a suitable formulation of the original problem, onstru tion of a orre t relaxation s heme, stability analysis, optimality onditions, approximation theory, and numeri al implementation.
4.3.a
Original problem
Throughout this se tion, a xed time interval
I :# [0; T℄
will be used in pla e of
.
As we want to fo us our attention rather to a method of relaxation than to optimal
ontrol problems themselves, we will restri t a bit the full generality and onsider our
optimal ontrol problem in a so alled Bolza form 35
ODE
(POC )
35
T . Minimize X ' ( t ; y ( t ) ; u ( t )) d t % ( y ( T )) 6 6 6 0 6 6 6 dy 6 6 subje t to # f(t ; y(t); u(t)) : t ò I; 6 6 6 dt
a.a.
> 6 6 6 6 6 6 6 6 6 6 6 F
( ost fun tional) (state equation)
y(0) # y0 ; (initial ondition) ( t ; y ( t ) ; u ( t )) ¢ 0 :a.a. t ò I ; (state ontrol onstraints) u(t) ò S(t) :a.a. t ò I; ( ontrol onstraints) y ò W 1 q (I; Rn ); u ò L p (I; Rm ); ;
This is a spe ial form of the Bolza problem on this xed time interval with a xed initial ondition.
In general, one an onsider
y0 as an additional ontrol variable and # (y(0); y(T)), and possibly t # 0 and T .
also additional state onstraints at time
Ë
278
4 Relaxation in Optimization Theory
' : I , Rn , Rm Ù R, f : I , Rn , Rm Ù Rn , y0 ò Rn , S : I Â± Rm a multivalued n Ù R, and mapping, : R : I , Rn ,Rm Ù R are subje ted to ertain data quali ation introdu ed later, n ; m £ 1, 1 ¢ p %, 1 q ¢ %. Of ourse, R is expe ted to be ordered by a one D so that the ondition ( t ; r ; s ) ¢ 0 has a sense.
where
This problem ts with the framework of Se tion 4.1 if one takes the data for the problem (POC ) as
Y # W 1 q (I; Rn ) ; U # L p (I; Rm ) ; Uad # {u ò U; :a.a. t ò I : u(t) ò S(t)} ; ;
(4.63a) (4.63b)
X # L q (I; Rn ) , Rn ;
(4.63 )
# L p (I; R ); D # { ò ; : t ò I : (t) ò D } ; dy (u ; y) # " Nf (y; u) ; y(0) " y0 ; dt B(u ; y) # N (y; u) ;
(4.63d)
J(u ; y) #
T X ' ( t ; y ( t ) ; u ( t )) d t % ( y ( T ))
(4.63e) (4.63f)
:
(4.63g)
0
u and then
For optimality onditions, we will onne ourselves to
independent of
onsider
has a nonempty interior in
# C(I; R ). Let us note that then the one D C(I; R ) provided D has a nonempty interior in R .
Example 4.32 (Nonexisten e of optimal ontrols: os illations).36A very simple and illustrative problem whi h orrupts existen e of solutions is:
T
Minimize
J(y; u) :# X (u(t)2 "1)2 % y(t)2 dt
( ost fun tional)
0
subje t to
dy # u; y(0) # 0; dt y ò W (I); u ò L (I): 1;4
As
J
(4.64)
( ontrolled system)
4
is nonnegative, the inmum of (4.64) must be nonnegative, too. A tually, this
inmum is zero. The minimizing sequen e of ontrols is, for example,
u " (t) # Then
1 "1
t " ¡0 otherwise : if sin( / )
(4.65)
J(u " ; y " ) # O("2 ). Yet, there is no ontrol su h that J(u ; y) # 0 for ddt y # u and T
T
y(0) # 0. Indeed, then both P0 (u(t)2 "1)2 dt # 0 and P0 y(t)2 dt # 0, so that y # 0, and from
36
d dt y
T
# u we have also u # 0, whi h however ontradi ts P (u(t) "1) dt # T #Ö 0. 2
2
0
This lassi al ounterexample is essentially due to Bolza [129℄; f. also Ioe and Tikhomirov [399,
Se t. 9.1.1℄. A similar example using the ost fun tional Se t. 61℄.
P
T
0
1%y
(
2
1 % (u "1) ) dt is by Young [808,
)(
2
2
4.3 Optimal ontrol of nitedimensional dynami al systems
Example 4.33 (Illustration of the nonexisten e due to os illations).
repla ements
Ë 279
Let us illustrate
the phenomenon from Example 4.32 on a simple ele tri al ir uit in Figure 4.2.
R heat/light onve tion ( oe ient
a2 )
R
( oe ient
i
u
heat onve tion
a1 )
umax
T
Fig. 4.2:
B
A simple ele tri al ir uit to
ontrol temperature of a lamp lament;
T is a transistor, B is a battery, and R is a temperaturedependent
ib
resistor (a bulb).
y # y(t) of a lament in a lamp to be as lose yd # yd and simultaneously the heat energy
Our aim is to ontrol the temperature
as possible to the desired temperature
lost (i.e. undesired heat produ tion) on the transistor to be as small as possible. The
umax is supposed onstant. Let the ontrol variable37 be the olle toremitter voltage u # u ( t ). The (absolute) temperature y is governed by the nonlinear supply voltage
dierential equation des ribing the energy balan e in the lament
where
dy (u " u(t)) % a y(t) % a y(t) # max ; y(0) # y ; dt R(y(t)) 2
4
1
2
(4.66)
0
¡ 0 is the heat apa ity (per unit length)
of the heated lament,
a1
and
a2
are the oe ients of the heat transfer via onve tion and radiation (due to the Stefan
R # R(r) is the temperaturedependent resistan e of y0 is the initial temperature of the lament. The sour e term on right2 hand side, namely ( u max " u ( t )) i ( t ) # ( u max " u ( t )) / R ( y ), is the Joule heat and i is the
olle tor urrent. The energy lost within a time interval I on the transistor is obviously T T P u ( t ) i ( t ) d t # P u ( t )( u max "u ( t ))/ R ( y ( t )) d t , hen e our problem is to minimize 0 0 Boltzmann law), respe tively, and the lament, and
J(u ; y) #
T u(t)(umax "u(t))/R(y(t)) X 0
%
y t " yd ) ( ( )
2
dt :
power lost on
deviation from the desired
the transistor
temperature of the lament
(4.67)
n # m # 1, '(t ; r; s) # r" yd )2 % s(umax "s)/R(r), $ 0, $ 0, f(t ; r; s) # "1 ((umax " s)2 /R(r) " a1 r " a2 r4 ), and S ( t ) # [0 ; u max ℄. Su h problem, however, has no solution, in general. Let us show 4 2 it on a spe ial ase y d ( t ) # y 0 for some y 0 ¡ 0 su h that a 1 y 0 % a 2 y 0 u max / R ( y 0 ). ODE Obviously, it ts with the problem (POC ) if one takes the data
(
37
In fa t, the olle toremitter voltage is itself ontrolled by the base urrent
a tual ontrol variable.
ib whi h is therefore the
280
Ë
4 Relaxation in Optimization Theory
Then it is possible to show that the ontrol
u k (t) #
uk ò L
I
( ) dened by
t ò [lT/k ; (l% a)T/k℄; l # 0; :::; k"1 ;
0
for
umax
elsewhere,
(4.68)
2 a # 1 " R(y0 )(a1 y0 % a2 y0 4 )u"max , drives the system arbitrarily near to y d # y 0 . More pre isely: for any " ¡ 0 one an nd k " ò N large enough so that for every k £ k " one gets ( u k ) " y d C I ¢ " , where ( u ) denotes the solution to (4.66). Therefore, the se ond term in (4.67) an be made arbitrarily lose to zero for y # ( u k ) while the rst one is identi ally zero for u # u k from (4.68). In other words, we showed
with
( )
that the inmum of su h problem is zero. Yet, if this inmum were a hieved, then
y # (u) would have to be identi ally equal to yd . By (4.66), it means that umax " u)2 # R(y0 )(a1 y0 % a2 y0 4 ). However, any ontrol u satisfying this requirement makes the rst term in the ost fun tional, i.e. u ( u max " u )/ R ( y 0 ), positive. This is a
ne essarily (
ontradi tion, showing that the inmum of our problem annot be a hieved.
Example 4.34 (Nonexisten e of optimal ontrol: on entration).
Another
phenome
non whi h an orrupt existen e of solutions an be demonstrated on the simple problem:
T
Minimize
J(y; u) :#X (2"2t% t2 )u(t) dt % (y(T)"1)2 0
subje t to
dy # u; y(0) # 0; dt y ò W (I); u ò L (I); u £ 0; 1;1
where
/
( ost fun tional) 7 7 7
7
( ontrolled system) ? 7 7
(4.69)
7 7 ( ontrol onstraint) G
1
T ¡ 1 is xed. If u ò L1 (I) would be an optimal ontrol, then u annot be identi
ally 0 (whi h would not obviously be optimal), and we an always take some part of this ontrol and add the orresponding area in a neighbourhood of 1. This does not ae t
T
y(T) but makes P0 (2 " 2t % t2 )u(t)dt lower, ontradi ting the optimality of the
original ontrol.38 The optimal ontrol has a hara ter of a so alled impulse ontrol, here meaning a Dira measure supported at
38
t # 1. The response on impulse ontrols is
a(t) :# t2 " 2t % 2 attains its minimum at the point t # 1 so that the optimal ontrol t # 1 provided T ¡ 1. Considering, for k ò bigger than 1/(T "1) and , the ontrol u k and the orresponding state y k given by
The oe ient
N
is for ed to on entrate around for ~
òR
u k (t) #
k~
0
if
t ò (1; 1%1/k) ;
otherwise
.
0
y k (t) # > k~(t"1) F
~
t ò (0; 1) t ò (1; 1 % 1/k) if t ò (1 % 1/ k ; T ) ; if
if
# ~ mintòI a(t) % (~"1) % O(1/k ). Sin e mintòI a(t) # 1 and that the sequen e {( y k ; u k )}kòN will minimize J provided ~ # 1/2; then obviously limkÙ J(y k ; u k ) # 3/4 #
then we an see that
J(y k ; u k )
2
2
inf J . On the other hand, this value inf J
annot be a hieved, i.e. the optimal ontrol does not exist.
This is here be ause of the on entration ee t. More pre isely, the sequen e { uniformly integrable.
u k }kòN
L (I) is not 1
Ë 281
4.3 Optimal ontrol of nitedimensional dynami al systems
typi ally dis ontinuous just at times when the ontrol is on entrated, whi h makes theory of su h ontrol systems very nontrivial.
Therefore, the need of relaxation appears very naturally even in a very simple situations. The reader an observe that the minimizing ontrol sequen es for Example 4.32
onverges weakly* to the Young measure
x
# Æ % Æ" 1
2
1
1
2
1
u
while the sequen e { k }
from (4.68) onverges weakly* in the sense of Young measures to the (relaxed) ontrol
x
# aÆ % (1" a)Æ umax . Both are, in fa t, unique optimal relaxed ontrols when speak0
ing in terms of Young measures. In real situations like in Example 4.33, su h 2atomi
hattering ontrol an be realized in pra ti e by fast os illating ordinary ontrols
É 10" " 10" 8
quite easily be ause the swit hingtime s ale of the transistor (
6
uk
se ) is
omparatively mu h shorter than the time s ale of the heating/ ooling pro ess of the
É 10" " 10"
lamp lament (
2
1
se ). This prin iple is a tually often used in the ontrol
te hnique, exploiting spe ial swit hing transistors spe ially designed to treat the on/o regimes.
Remark 4.35 (Original versus relaxed ontrols).
The reader may ask a question why
one needs the relaxed problem if one must eventually realize approximately the relaxed ontrols by the original ones? This relation reminds the relation between the differen e and the dierential equations the latter ones are an e ient analyti al tool to analyze a limit behaviour of the former ones. Here the aim of analysis of the relaxed problems is, beside purely theoreti al aspe ts, to establish some on rete properties of optimal relaxed ontrols, whi h may help to determine them or at least to get a theoreti al support for e ient numeri al methods. Moreover, the results valid for relaxed problems an usually be ree ted in appropriate results for the original problems, f. Corollaries 4.364.40.
Let us briey outline whi h sorts of results an be obtained for the original problems by analysing the relaxed problems. In parti ular, this an yield existen e of solutions to the original problem, the pointwise (Pontryagin's type) maximum prin iple for these solutions (if any), or some information about properties and behaviour of minimizing here
stands
asymptoti ally admissible sequen es for the original problem (PODE OC ); for the strong topology of C ( I ; R ). Moreover, by analysing the point
wise maximum prin iple for the relaxed problem, one an also get information about a limit behaviour of fast os illations of su h sequen es.39 To be more spe i , let us formulate a few outlined onsequen es pre isely; of
ourse, the reader is expe ted to read them again together with their proofs after going through Se tions 4.3.b, .
39
For example, if there is a unique optimal relaxed ontrol whi h is hattering, then every minimiz
ing sequen e must inevitably exhibit a unique pattern of fast os illations whi h tends to live in a neighbourhoods of parti ular atoms, f. Figure 3.3 for the ase of a twoatomi ontrol
# i H (u )% i H (u 1
2
1
1
2
2 ).
Ë
282
4 Relaxation in Optimization Theory
The natural basi data quali ation on erning
f
tives r and
'r
f
and
' and their partial deriva
are the following
f t ; r; s) ¢ a q (t) % b(r) % sp q ; /
(
p/ q
(4.70a)
f r (t ; r; s) ¢ a q (t) % b(r) % s
f r (t ; r1 ; s) " f r (t ; r2 ; s) ¢ (a q (t) % b(r1 ) % b(r2 ) % s
'(t ; r; s) ¢ a1 (t) % b(r) % s ;
' r (t ; r; s) ¢ a1 (t) % b(r) % s ;
' r (t ; r1 ; s) " ' r (t ; r2 ; s) ¢ (a1 (t) % b(r1 ) % b(r2 ) % s
;
(4.70b)
p/ q
r "r2 ;
) 1
p
(4.70d)
p
(4.70 )
(4.70e)
p
r "r2
) 1
(4.70f)
a1 ò L1 (I), a q ò L q (I), and b ò C(R% ) in reasing. To guarantee the existen e of the ontroltostate mapping : U Ù Y and the oer ivity of (PO ), we have to require additional spe ial quali ation, namely a lineargrowth of f ( t ;  ; s ) and the oer ivity of ' and with respe t to U ad , i.e.
with some
; a ò L q (I) ; ò R% : f(t ; r; s) ¢ (a (t) % sp q )(1 % r); ; a ò L (I) ; b ò R% : t ò I : r ò Rn : s ò S(t) : '(t ; r; s) £ a(t) % bsp and inf ¡ " : /
1
1
1
(4.70g)
1
1
(4.70h)
The maximum prin iple will involve the Hamiltonian40 given by41
h y 0 ;
*
;
(
*
t ; s) # (t)  f(t ; y(t); s) " 0 '(t ; y(t); s) : *
*
(4.71)
Corollary 4.36 (Maximum prin iple for (PODE OC )).42 Let p ò [1 ; %), q ò (1 ; %), the one ODE D R has a nonempty interior, (POC ) possesses an optimal solution (y; u) su h that
(
y; i H (u))
40
ODE
solves the relaxed problem (R H P
OC
)
introdu ed later,43
independent of
s
Sometimes, the expression in (4.71) is alled pseudoHamiltonian or Pontryagin's Hamiltonian.
h (t ; r; s* ) # supsòS(t) (s*  f(t ; r; s) " '(t ; r; s)); f. h (t ; y(t); *(t)) # h Sy;1; (t) dened here by (4.75), provided *0 # 1. In fa t, we will derive the Hamiltonian (4.71) up to an integrable onstant (dependent on t ). This
Also, the Hamiltonian is sometimes dened rather as Clarke [223℄. Then obviously
41
*
does not inuen e the maximum prin iple (4.75), only it would ae t Remark 4.42.
42
The formulae (4.72), (4.74) and (4.75) represent a very lassi al version of the pointwise maxi
mum prin iple ex ept the onstan y of the Hamiltonian in time along the optimal pair (
y; u), f. Re
mark 4.42, whi h is irrelevant in our theory be ause the Hamiltonian resulting by our derivation is determined uniquely only up to integrable fun tions of time. Beside the original works by Boltyanski , Gamkrelidze, and Pontryagin [127, 616℄ generalizing Hestenes [382℄ and the monograph by Pontryagin, Boltyanski , Gamkrelidze and Mish henko [617℄, we refer also to Balakrishnan [49℄, Barbu [76℄, Berkowitz [110℄ and Medhin [113℄, Boltyanski and Poznyak [128℄, Cesari [196℄, Clarke [222℄, Colonius [236℄, Gabasov and Kirillova [340℄, Hartl, Sethi and Vi kson [377℄, Hestenes [384℄, Ioe and Tikhomirov [399, Se t.2.4℄, Kaskosz [420℄, MagarilIl'yaev [509℄, Mordukhovi h [550℄, Neustadt [574℄, Zeidler [812℄, et .
43
This just meas that there is no relaxation gap, i.e.
ODE min(PODE OC ) # min(RH POC ).
It happens if the
problem is value Hadamard wellposed with respe t to suitable perturbations ( f. Remark 4.7) in
Ë 283
4.3 Optimal ontrol of nitedimensional dynami al systems
ò C(I , Rn ; Rk,n ) with r (t ; r) # (t ; r)/r, ò C(Rn ; Rn ), and (4.70) be valid. Then there are £ 0 and ò r a(I; R ) su h that ( ; ) #Ö 0, £ 0, the
and
r
*
*
*
0
*
*
*
0
omplementarity ondition
y℄  # 0 *
[
on
I
(4.72)
is valid, and the integral maximum prin iple
T X h y; 0 ; *
0
(
*
t ; u(t)) dt # sup
state
ò Lq
(
I; R
*
u ò U ad
(
*
0
t ; u (t)) dt
(4.73)
R
h y; 0 ; òCarp (I; m ) is dened by (4.71) with the adjoint n ) solving44 the ba kward terminalvalue problem:
is valid, where the Hamiltonian *
T X h y; 0 ;
*
*
d % f r (t ; y(t); u(t)) # ' r (t ; y(t); u(t)) % dt (T) # " (y(T)) : *
*
*
* ; /
r ( t ; y ( t ))
0
*
(4.74)
? G
S is measurable losedvalued, and if S(t) is bounded in Rm uniformly with respe t to t or 0 ¡ 0,45 then also the pointwise maximum prin iple is valid:46
Moreover, if
*
:
a.a.
tòI :
h y 0 ;
*
;
(
*
t ; u(t)) # max h y 0 ;
sòS(t)
*
;
(
*
t ; s) :
(4.75)
Proof. The formulae (4.73), (4.74) and (4.75) are respe tively just (4.104), (4.105) and (4.106) below for
# i H (u),
so that the assertion follows dire tly from Proposi
ODE tion 4.50 for the relaxed problem (R POC ) with a suitable separable H
H Carp (I; Rm )
whose existen e is guaranteed by the data quali ation (4.70af); f. Example 4.56. Moreover, the abstra t omplementarity (4.23a) yields the integral omplementarity
T
t ; y(t))  (dt) # 0 from whi h the lo al omplementarity (4.72) realizing that y ¢ 0 and £ 0 everywhere on I . P
0
*
(
*
follows when
Å
*
Corollary 4.37 (Maximum prin iple for minimizing sequen es).47 Let p ò [1; %), q ò (1 ; %), and (4.70ad,fh) be valid, while (4.70e) be strengthened to ' r ( t ; r; s ) ¢ a (t) % b(r) % sp with some ¡ 1, let # 0 (i.e. there are no state onstraints) and
/
1
let {( u k ;
y k )}kòN be a minimizing sequen e for (PODE OC ). Then T
:u ò Uad : lim inf X h y k k Ù
0
;1;
k ( t ; u k ( t )) *
parti ular, if there are no state onstraints (i.e.
" h yk
;1;
k ( t ; u ( t )) d t *
£ 0;
(4.76)
# 0) or if the problem has a linear/ onvex stru ture
( f. Example 4.57).
44
Of ourse, sin e
* is a measure, (4.74) is to be understood in the sense of distributions. Then, from d * n ) so that, in fa t, *ò BV( I ; n ). dt ò r a( I ;
(4.74) one an read that
R
45
The latter ondition applies, in parti ular, if
46
Let us note that, for a.a.
example, at
47
s # u(t).
# 0.
R
t ò I , the maximum on the righthand side of (4.75) is a tually attained, for
Optimality onditions for minimizing sequen es have been also investigated by Medhin [532℄, Po
lak and Wardi [615℄, Sumin [738℄, Hamel [375℄ et ., f. also Sumin [739℄ for paraboli optimal ontrol problems.
Ë
284
4 Relaxation in Optimization Theory
h y; 0 ; is given again by (4.71) while the adjoint state n ) solves the ba kward terminalvalue problem
where the Hamiltonian
W min(q; ) (I; 1;
R
*
*
d k % f r (t ; y k (t); u k (t)) k # ' r (t ; y k (t); u k (t)) ; dt *
*
k (T) # 0 :
*
k ò *
(4.77)
Proof. The assertion follows from Proposition 4.50. Indeed, let us make a relaxation (R
ODE
H POC
) by a suitable separable
not hold, we get some {(
u k ; y k )} su h that
lim
k Ù
u ò Uad
T X h y k ;1; 0
H;
f. Example 4.56. Supposing that (4.76) does
and a subsequen e, denoted for simpli ity again by
*
k
(
t ; u k (t)) " h y k
u
By the oer ivity (4.70h), { k } is bounded in
;1;
k ( t ; u ( t )) d t *
0:
(4.78)
L p (I; Rm ), {y k } is bounded in W 1 q (I; Rn ), ;
W min(q; ) (I;
R
n ) so that we an suppose that and eventually also { } is bounded in k * 1; q i H (u k ) Ù weakly* in H , y k Ù y weakly in W (I; n ), and also *k Ù * weakly 1 ; min( q ; ) n ). Sin e {( u ; y )} is minimizing, by Proposition 4.46 the limit ( ; y ) in W (I; k k *
solves (R
(
R
R
) and, passing to the limit in (4.77), we an also see that
*
solves (4.105)
0 # 1 and # 0. Realizing that y k Ù y and k Ù also in the norm of I; Rn ), we an show that h y k 1 k Ù h y 1 in the norm of Carp (I; Rm ); f. Exam
with
L
ODE
H POC
1;
*
*
*
;
;
*
;
;
*
*
ple 3.106. Therefore
lim
k Ù
T X 0
h yk
;1;
k ( t ; u k ( t )) *
" h yk
;1;
k ( t ; u ( t )) d t *
# lim i H (u k ) " i H (u); h y k k Ù
;1;
k *
# " i H ( u ) ; h y
By (4.104), this limit annot be negative; realize that no other
;1;
*
: *
satisfying (4.105) does
exist. This gives the sought ontradi tion with (4.78).
Å
Corollary 4.38 (Non on entration of minimizing sequen es). Let p ò [1; %), q ò (1 ; %℄, f satisfy (4.70a,b,g), ' satisfy (4.70d,e,h), and ò C(I ,Rn ; R ). If {(u k ; y k )}kòN asymptoti ally admissible sequen e for (PODE OC ), then the ontrols do not p 1
on entrate energy, i.e. { u k ; k ò N} is relatively weakly ompa t in L ( I ).
is a minimizing
Proof. This assertion is just the onsequen e of Propositions 4.46(iiiiv) and 3.79
H . It is imporH does exist; f. Example 4.56 p with the modi ation that H an also ontain a (separable) subspa e C ( I ) Ô (R ) with 0 m some omplete separable subring R C (R ) to satisfy the assumptions of Proposiwhi h uses a relaxation by a su iently ri h but separable subspa e tant that, for given
' and f
satisfying (4.70a,b,d,e), su h
tion 3.79.
On spe ial o
asions, the relaxed problem may serve to establish existen e of solutions to the original problem. Let us just illustrate su h sort of results obtainable by two ompletely dierent te hniques: either by a suitable onstru tion of a 1atomi solution from an arbitrary (or at least some) relaxed optimal ontrol as used in the proof
4.3 Optimal ontrol of nitedimensional dynami al systems
Ë 285
of the FilippovRoxin theorem 4.29 or by usage of Bauer's prin iple (Theorem 1.13) together with a hara terization of extreme points48:
Corollary 4.39 (Existen e of solutions to (PODE OC )). (4.63b) be nonempty with
S
Let
p ò [1; %), q ò (1; %℄, Uad from
measurable and losedvalued, (4.70a,b,d,e,g,h) be valid,
and at least one from the following sets of onditions on
:
a.a.
t ò I : r ò Rn :
', f , S and
be satised:
the orientor eld
Q(t ; r) # '(t ; r; s)% R%0 ; f(t ; r; s) ;
(
t ; r; s)% D ò R1%n% ; s ò S(t)
is losed onvex
'(t ; r; s) # ' 0 (t ; r) % ' 1 (t ; s) with ' 0 (t ; ) on ave; f(t ; r; s) # f 0 (t ; r) % f 1 (t ; s) with f 0 (t ; ) ane; S(t) is bounded (uniformly in t) and satises (3.30); # 0 ; i.e. no state onstraints :
or
(
(
)
(
)
(
)
(
)
(
)
)
(4.79)
/ 7 7 7
(4.80)
? 7 7 7 G
ODE
Then the original problem (POC ) has a solution. Proof. As to the rst option (4.79), it su es to verify the onditions of Theorem 4.29
and with A being the linear mapping y ÜÙ ddt y while, for the purpose of this proof, the initial ondition y (0) # y 0 may be involved as a state onstraint.
with
I
in pla e of
As to the se ond option (4.80), it just su es to realize that the relaxed prob
H L1 (I; C0 (Rm )) onsists in minimization of a on ave ost fun tional over the onvex weakly* ompa t set U . By Bauer's extremal prin iple (Theoad ODE lem (R POC ) with H
rem 1.13), su h problem admits at least one extreme solution. Using the ane home
N M : Y(I; S0 ) Ù U ad with M and S0 from (3.30), we an see that the points of U ad pre isely orresponds to the extreme points of Y(I; S0 ) whi h
omorphism extreme
are, by Proposition 3.9, a.e. just Dira distributions. Therefore, this extreme solution must be again 1atomi . We shall onsider also a perturbed problem depending on perturbation parameters
"1 ; "2 ¡ 0:
ODE
(POC ; "
48
1 ; "2
)
. Minimize 6 6 6 6 6 6 6 6 6 subje t to > 6 6 6 6 6 6 6 6 6 F
T " " X ' 1 ( t ; y ( t ) ; u ( t )) d t % 1 ( y ( T )) 0
dy # f "2 (t ; y(t); u(t)) ; y(0) # y "2 ; dt " 2 ( t ; y ( t )) ¢ " 1 ( t ) for all t ò I ; u(t) ò S(t) (:a.a. t ò I ) ; q n y ò W (I; R ); u ò L p (I; Rm ): 0
1;
For usage of this te hnique even to more ompli ated situations we refer to Balder [56℄, Cellina
and Colombo [191℄, Cesari [195, Chap. 16℄, Mari onda [518℄, or Raymond [629633℄. For problems with linear ost fun tions see Neustadt [573℄ and Ole h [581℄, or also Gabasov and Kirillova [340, Se t. V.3℄.
286
Ë
4 Relaxation in Optimization Theory
The perturbed data are to approximate the original ones in the following sense:
with some
f " (t ; r; s) " f(t ; r; s) ¢ (a0 (t) % b0 (r) % 0 sp )" ; "
(4.81a)
p
' (t ; r; s) " '(t ; r; s) ¢ (a0 (t) % b0 (r) % 0 s )" ; "
(r) " (r) ¢ b0 (r)" ; "
y0 " y0 ¢ " ;
"
(
t ; r) "
(
(4.81b)
t ; r) ¢ " ;
(4.81 )
"
(t) ¢ "
(4.81d)
a0 ò L1 (I), b0 : R% Ù R% ontinuous in reasing, and 0 ò R% .
Corollary 4.40 (Stability of minimizing sequen es for (PODE ò [1; %), OC )). Let p q ò (1; %℄, f and f " satisfy (4.70a,b,g), ' and ' " satisfy (4.70d,e,h), ; " ò C(I , Rn ; R ), " ò C(I; R ), " (t) ¡ 0 for all t ò I , and (4.81) be satised. % % Then there is E : R Ù R su h that lim
" 1 ; " 2 Ù0 "2 ¢E("1)
ODE inf (PODE OC " 1 " 2 ) # inf (POC ) : ;
(4.82)
;
R% , R% with " k # (" k ; " k ) ¢ E(" k ) with E : R% Ù R% guaranteeing
Moreover, let a positive nonin reasing sequen e { " k } k òN
1;
2;
0; 0) be given su h that " k " (4.82) and let { u } k òN with " # ( " ; " ) be a minimizing asymptoti ally admissible k ODE sequen e for (POC " " ). Then there is an in reasing fun tion : N Ù N su h that any 1 2 "n sequen e { u } n òN with k n £ ( n ) is a minimizing asymptoti ally admissible sequen e kn
onverging to (
2;
1
;
1;
2
;
ODE
for (POC ). Proof. This assertion is a onsequen e of Corollary 4.6 and Proposition 4.47 if one realizes that for a ountable family of optimization problems a relaxation by a ommon separable
H does exist, f. Example 4.56 below. The separability of H ensures metrizH , as required in
ability of the relativized weak* topology on bounded subsets of
*
Å
Corollary 4.6.
Remark 4.41 (Lagrange and Mayer problems). Spe ial ases of the Bolzatype probODE lem (POC ) are when # 0 ( alled the Lagrange problem) or ' # 0 ( alled the Mayer problem). The Bolza problem looks most general but, in fa t, both the Lagrange and the Mayer forms are of the same power at least if they are no distributed state onODE straints and initial state is ontrolled, too.49 In parti ular, any Lagrange problem (POC )
# 0 an be transformed into the Mayer problem by inventing an auxiliary state d dt y n% # f(t ; y ; :::; y n ; u) with the initial ondition y n % (0) # 0, and then onsidering the terminal ost fun tional y n % ( T ).
with
y n%1
and the additional dierential equation
1
1
1
1
The mentioned ontrol of the initial onditions would lead to a fully general Bolza problem involving the ostfun tional term
49
1 (y(0); y(T)).
Transformations between these lasses of problems are thoroughly treated, e.g., in the lassi al
monograph by Cesari [196℄.
Ë 287
4.3 Optimal ontrol of nitedimensional dynami al systems
Remark 4.42 (Constan y of the Hamiltonian along optimal traje tories). re
ondition
h y 0 ;
*
;
(
*
is
t ; u(t))
sometimes
the
maximum
is onstant in time for any optimal pair (
autonomous systems, i.e.
prin iple,
Still one monamely
that
u ; y). This a tually holds for
', f , and S independent of time and in the un onstrained smooth in the s variable 0
$ 0 and # 1. Assuming ' and f *
ase. In parti ular, i.e. and
ompleting
S onvex, (4.75) gives
(t)  f s (t ; y(t); u(t)) # ' s (t ; y(t); u(t)) % N S (u(t)) : *
(4.83)
t
Then, by the following (formal) al ulations (with the variable not expli itly written), we have
d d h (t ; u(t)) #  f ( y; u ) %  f ( y; u ) " ' ( y; u ) t t dt y dt dy du %  f r (y; u) " ' r (y; u) %  f s (y; u) " ' s (y; u) dt dt ò NS u t #  f t (y; u) " ' t (y; u) ; (4.84) *
*
*
;
*
*
*
0
*
(
( )) by (4.83)
d dt y
# f(y; u) and also the adjoint equation (4.74), together with # 0 for a.a. t ò I . From this we an see that h y (t ; u(t)) is onstant in time if both f t # 0 and ' t # 0. For the state onstraint #Ö 0, the Hamiltonian
where we used
N S (u(t)) ddt u(t)
h y 0 ;
*
;
; *
(
*
;
*
t ; u(t)) is not onstant in time. In fa t, to obtain su h additional ondition,
one must augment the Hamiltonian (4.71) as
h y 0 ;
*
;
; *
(
*
t ; s) # (t)  f(t ; y(t); s) " (t) *
*
(
t ; y(t)) " 0 '(t ; y(t); s) : *
(4.85)
This does not ae t the maximum prin iples (4.73) and (4.75) themselves be ause
s but allows us to enhan e (4.84) to obtain ddt h y 0 (t ; u(t)) #  f t (y; u) "  t (y) " 0 ' t (y; u). In parti ular, for the autonomous system and the original Hamiltonian from (4.71), we obtain h y ( t ; u ( t )) # ( t ) ( t ; y ( t )) up to 0 does not depend on *
*
;
*
*
;
*
;
*
;
*
;
*
;
*
*
a fun tion onstant in time. Exe uting this al ulus dire tly for the relaxed problem below, we oud avoid smoothness in
svariable and onvexity of S.
Remark 4.43 (Timeoptimal ontrol).
Some problems uses the terminal time
ontrol variable and the ost fun tional just equal to
T
as a
T . When onsidering the terminal
onstraints as in Remark 4.41, we obtain a timeoptimal ontrol problem. Now the data
f : [0; %) , Rn , Rm Ù Rm m and S : [0 ; %) Â± R . When res aling time to a xed interval, say [0 ; 1℄, su h problems an be transformed into a Mayertype problem with T n a position of a s alar is to be dened on not apriori bounded time intervals, i.e.
ontrol parameter: Minimize subje t to
y n%1 (1) dy i Tf (tT; y(t); u(t)) ; y i (0) # y0i for i # 1; :::; n ; # i dt T for i # n %1 ; ( y (1)) ¢ 0 for all t ò [0 ; 1℄ ; u(t) ò S(tT) (:a.a. t ò [0 ; 1℄) ; y ò W 1 q (0; 1; Rn%1 ); u ò L p (0; 1; Rm ); T £ 0: ;
/ 7 7 7 7 7 7 7 7 ? 7 7 7 7 7 7 7 7 G
(4.86)
Ë
288
4 Relaxation in Optimization Theory
Remark 4.44 (Multi riteria problems). the multiplier
0
*
is a ve tor.
' and ve torvalued, and
One an onsider
Even for
'
and
s alarvalued, one an onsider a
ODE multi riteria modi ation of (POC ) to minimize (in a Pareto or Slater sense) both
P
T
0
'(t ; y(t); u(t)) dt
4.3.b
and
(y(T)), and then 0 ò R2 . *
Relaxation s heme, orre tness, wellposedness
ODE Using the previously developed theory, we will make the relaxation of (POC ) by means
of a suitable
C(I)invariant
subspa e
H
of
Carp (I; Rm ). Throughout this se tion, we
will suppose (without any loss of generality)
H
to be a normed linear spa e with a
topology ner than the natural topology oming from
Carp (I; Rm ). We take Uad from
(4.63b) and, likewise we did in (4.30), we put
U ad # b lH with
Uad
B
*
;
B i H ( U ad )
YHp (I; Rm ) L p (I; Rm ).
denoting, of ourse, the norm bornology on
is de omposable in the sense (4.41) so that
U ad
(4.87) Let us note that
is always onvex; f. also Re
mark 3.13. Furthermore, we will assume that the twoargument Nemytski mapping
f
N f : W 1; q (I; Rn ) , L p (I; Rm ) Ù L p (I; Rn ) admits a ontinuous extension N : p W 1; q (I; Rn ) , Y H (I; Rn ) Ù L q (I; Rn ) and extend the original initial value problem
to
f
dy/dt # N (y; )
with
y(0) # y0 .
Moreover, we will onne ourselves to the
ase when the extended Nemytski mapping is ane with respe t to the (relaxed)
ontrol, namely when it takes the form
W 1 q (I; Rn ) L ;
(
f
N (y; ) # f y Ǳ ;
see Lemma 3.101. Sin e
I; Rn ), this semiane extension will ertainly exist if f ò CAR H
;
p; q
(
I , Rn , Rm ; Rn ) :
(4.88)
Then the extended initialvalue problem takes the form
dy # f y Ǳ ; y(0) # y : dt
(4.89)
0
In other words, we extend
L q (I; Rn ) , Rn
from (4.63e) to
: YH (I; Rm ) , W 1 q (I; Rn ) Ù p
;
dened by
( ; y) #
dy " f y Ǳ ; y(0) " y dt
0
:
(4.90)
Analogously, supposing
' òCAR H
;
for some
p1
p;1
(
¡ 1,
I , Rn , Rm ; R)
and
òCAR H
;
p; p1
(
I , Rn , Rm ; R )
we an extend also the Nemytski mapping
in the ost fun tional, so that the ost fun tional
J
N'
(4.91) appearing
from (4.63g) extends to
4.3 Optimal ontrol of nitedimensional dynami al systems
Ë 289
J : YH (I; Rm ) , W 1 q (I; Rn ) Ù R dened by p
;
T X ['
J ( ; y) #
0
y Ǳ ℄(dt) % (y(T)) ;
(4.92)
' y Ǳ is to be understood possibly in the sense of measures on I . And also the p mapping B from (4.63f) is to be extended analogously, being valued in L 1 ( I ; R ). Thus
where
we ome to the following relaxed problem:
(R
ODE
H POC
T X ['
. Minimize 6 6 6 6 6 6 6 )
0
y Ǳ ℄(dt) % (y(T))
dy # f y Ǳ ; y(0) # y ; dt y Ǳ ¢ 0 a.e. on I; p ò U ad YH (I; Rm ) ; y ò W
subje t to > 6 6 6 6 6 6 6
F
0
1;
q (I;
Rn
)
:
Lemma 4.45 (Corre tness of the extended state problem). Let p ò [1; %) and q ò p m (1 ; %℄, let H be a C ( I )invariant separable subspa e of Car ( I ; R ), f satisfy (4.88) and additionally50 also the growth ondition (4.70g). Then: The extended state equation (4.89) possesses for any
(i)
y # () ò W 1 q (I; Rn ). p m 1 q n The mapping : YH ( I ; R ) Ù W ( I ; R ) thus p m %
ontinuous if restri ted on YH % ( I ; R ) with any % ò R .
p
;
tion (ii)
ò YH (I; Rm ) a unique solu
;
dened is (weak*,weak)
;
# i H (u) with u ò L p (I; Rm ), then y # () solves the original initialvalue probODE p m 1 q n lem in (POC ). In other words, i H # where : L ( I ; R ) Ù W ( I ; R ) denotes
(iii) If
;
the original ontroltostate mapping.
ò YH (I; Rm ), there is a sequen e {u k }kòN bounded in L p (I; Rm ) su h that i H (u k ) Ù weakly* in H . To prove the existen e of the solution to (4.89), we shall just pass to the limit with the solutions y k that orresponds to u k , Proof. By the very denition of
p
*
whi h means
dy k # f(t ; y k ; u k ) ; y (0) # y : (4.93) dt p n By the lassi al theory, (4.70g) ensures for any u k ò L ( I ; R ) the existen e of just q n one solution y k ò W ( I ; R ); see Proposition 1.36. Besides, f ( t ; r; u k ( t )) satises the 0
1;
growth ondition
f t ; r; u k (t)) ¢ (a1 (t) % 1 u k (t)p
/
(
a k # a1 % 1 u k p p m bounded in L ( I ; R ).
with is
50
/
q
1 % r) ¢ a k (t)(1 % r)
)(
(4.94)
q bounded in Therefore,
L q (I) independently of k be ause {u k }kòN t y k ( t ) # y 0 % P f ( ; y k ( ) ; u k ( )) d ¢ y 0 % 0
Let us note that (3.192b) with Remark 3.104 turns out here to (4.70a), whi h would not guarantee the
existen e of the solution of our initialvalue state problem, however. For this reason we must impose the stronger growth ondition (4.70g).
Ë
290
4 Relaxation in Optimization Theory
t
a k ()(1 % y k ()) d, whi h shows via the Gronwall inequality that {y k }kòN is L (I; Rn ). Then, from (4.93) with (4.94), we an also see that {y } ò is 1 q n bounded even in W ( I ; R ). Hen e, taking possibly a subsequen e (denoted, for sim
P
0
bounded in
;
pli ity, by the same index), we an suppose that
yk Ù y
weakly (or, for
q # %, weakly*) in W 1 q (I; Rn ) : ;
(4.95)
Let us now pass to the limit in (4.93). The lefthand side obviously onverges to
dy/dt thanks to (4.95). The righthand side f(t ; y k ; u k ) an be written in the form f y k Ǳ k for k # i H (u k ). Realizing that (4.95) implies y k Ù y strongly in L (I; Rn ), we q n
an use Lemma 3.101 to obtain f y k Ǳ k Ù f y Ǳ weakly in L ( I ; R ). This shows q n that y ò W ( I ; R ) from (4.95) satises d y/d t # f y Ǳ . q ( I ; R n ), hen e Also, y (0) # y be ause y k (0) # y and y k Ù y weakly in W n also strongly in C ( I ; R ), and in parti ular y k (0) Ù y (0). Altogether, the existen e of
1;
0
1;
0
a solution to (4.89) has been demonstrated. Now, we are to prove the uniqueness of this solution. Supposing
y1 ; y2
are two
solutions to (4.89), we have
d( y " y dt 1
Then, supposing
!! !! y 1 ( t )
# (f y " f y ) Ǳ : 1
" y (t)!!!! # 2
h p (t ; s) # sp .
!! t !!X [( f !! ! 0 t X a2
!!
y " f y ) Ǳ ℄() d!!!! 1
2
% b (y 2
0
Realizing that
y1
1 )
and
taking into a
ount the initial onditions for all
tòI
!
% b (y 2
y2
2 )
% h p Ǳ y "y d 2
1
2
are apriori bounded in
L
I; Rn ) and y1 (t) # y2 (t)
y1 (0) # y0 # y2 (0), one gets
(
as a onsequen e of the Gronwall inequality generalized by a ontinuous
extension for the ase naturally
2
h p ò H , one gets by (3.192 )51
¢ with
2)
y # ().
h p Ǳ òr a(I). The point (i) has thus been demonstrated, putting
restri ted on YH % (I; Rm ) with % ò R% arbitrary. p m Let us take a sequen e { k } k òN Y H % ( I ; R ) su h that k Ù weakly* in H , and denote y k # ( k ). In other words,
Now we will show the ontinuity of
p
;
*
;
dy k # f y k Ǳ k ; y k (0) # y : (4.96) dt As previously, supposing h p q ò H , we an obtain the estimate d y k /d t ¢ (a % h p q Ǳ k )(1 % y k ). Again, we an dedu e that y k L I Rn ¢ C( a % 0
/
1
51
1
/
f t ; r1 ; s)
Let us note that (3.192 ) turns out here to (
2 sp )r1 " r2 with some a2 ò L1 (I), b2 : %. and 2 ò
R
R% Ù R%
( ;
" f(t ; r ; s) ¢ (a (t) % b (r 2
2
1
)
2
1 )
% b (r 2
2 )
%
arbitrary ontinuous in reasing ( f. Remark 3.104)
4.3 Optimal ontrol of nitedimensional dynami al systems
Ë 291
1 h p q Ǳ k L1 I ) with a suitable C : R% Ù R% , and then also dy k /dt L q I;Rn ¢ a1 %
1 h p q Ǳ k L q I C( a1 % 1 h p q Ǳ k L1 I ). As the sequen e {h p q Ǳ k }kòN is bounded in p L q (I) if k ranges YH % (I; Rm ), we an dedu e as previously that the sequen e {y k }kòN 1 q n is bounded in W ( I ; R ), and then we an take a weakly onvergent (possibly sub) sequen e. Let us denote its limit by y . Now our only task is to show that y # ( ), but /
( )
/
( )
(
/
)
/
( )
;
;
the limit passage in (4.96) is entirely the same as performed previously for the spe ial
k # i H (u k ). By the already proved uniqueness of the solution to (4.89), even the whole sequen e { y k } k òN onverges to y . This ompletes the proof of the point (ii).
ase
The last fa t to prove, namely the point (iii), follows immediately by Lemma 4.11.
Proposition 4.46 (Corre tness of the relaxation s heme). Let H be a C(I)invariant Carp (I; Rm ), p ò [1; %), q ò (1; %℄, (4.70g), (4.88) and (4.91) be valid, ò C(I , Rn ; R ), (PODE OC ) admits a bounded asymptoti ally admissible se
separable subspa e of
quen e,52 and (i)
(R
H
' and be oer ive in the sense (4.70h). Then: pnon on entrating.
ODE POC ) has a solution, and every solution is ODE
(ii) Every solution to (R P
H
OC
)
asymptoti ally admis
an be attained by a minimizing
ODE sible sequen e for (POC ).
(iii) Conversely, a limit of every minimizing
asymptoti ally
admissible weakly* on
verging sequen e for (POC ) (when embedded via i H ) solves (R H P
ODE
ODE
OC
).
Proof. First, let us noti e that Lemma 3.101 and the ompa tness of the embedding
W 1 q (I; Rn ) L
I; Rn ) guarantee that J ( ; y) : YH (I; Rm ) , W 1 q (I; Rn ) Ù R, dened by (4.92), is the weakly* ontinuous extension of the original ost fun tional J from ;
p
(
;
(4.63g). To verify the oer ivity ondition (4.21), let us just estimate, by (4.70h),
(u) # J(u ; (u)) £
T X a(t) dt 0
% b u pLp I Rm % inf (Rm ) ; ( ;
)
(u) Ù % for u L p I;Rm Ù % and u ò Uad . p m Then the level sets of are ontained in some Y H % (I; R ) H
from whi h we get
(
)
;
p priately large. We an use the weak* ompa tness of U YH ; % ( I ; ad
with
% ò R appro
and R with R(u) # B(u ; (u)); () # J ( ; ()) and R () # B ( ; ()). Then one
weak* ontinuity of all involved mappings, i.e. both see Lemma 4.45 and realize that
R
*
m ) together with the
gets the points (i)(iii) by using Proposition 4.1. As to the non on entration laimed in (i), let us suppose that the optimal relaxed
is not pnon on entrating. Then it diers from its pnon on entrating mod i ation whi h does exist thanks to the separability of H , see Proposition 3.81. By Lemma 4.23, ò U ad and drives the ontrolled system to the same state y as the ontrol be ause f , having the p/ q growth, has a growth lesser than p sin e q ¡ 1. Yet,
ontrol
52
Re all that throughout the whole se tion
refers to the strong topology of C(I;
R
).
Ë
292
4 Relaxation in Optimization Theory
by Lemma 4.22(ii) and (4.70h),
a hieves lower ost than the ontrol
whi h thus
Å
annot be optimal, a ontradi tion.
Further natural question on erns stability of the relaxed problem to the perODE turbed problem (POC ; "
1 ; "2
), denoted naturally as
min(RH PODE OC " 1 " 2 ). ;
;
Proposition 4.47 (Stability of relaxed problem). Let H Car (I; Rm ) be C(I)invariant, pq p n m n " n m ( I,R ,R ; R), p ò [1; %), q ò (1; %℄, f " ; f òCARH ( I,R ,R ; R ), ' ; ' òCAR H " ; ò C ( I , R n ; R ), " ò C ( I ; R ), " ( t ) ¡ 0 for all t ò I (in parti ular, the one D R p
;
;
;
;1
f " uniformly with respe t " ¡ 0, the oer ivity ondition (4.70h) be fullled both for ' and for ' " , and (4.81)
must have a nonempty interior), (4.70g) be valid both for to
hold. Then the relaxed perturbed problem (R H P there is
E:R ÙR %
%
f
and
ODE OC ;
" 1 ; " 2 ) always possesses a solution and
su h that
lim
" 1 ; " 2 Ù0 "2 ¢E("1)
min(RH PODE"1 "2 ) # min(RH PODE ) ; OC ;
;
(4.97)
OC
Limsup Argmin(RH PODE"1 "2 ) Argmin(RH PODE ) : OC ;
" 1 ; " 2 Ù0 "2 ¢E("1)
;
(4.98)
OC
ODE
Sket h of the proof. The fa t that (R H POC ; " ; " ) has a solution follows simply from Propo1 2 sition 4.46. Then the stability (4.97) and (4.98) follow readily from Proposition 4.5 modied for the ase
" repla ed by "1
;
"2 (u)
T
# P ' "1 (t ; y "2 ; u) dt with y " from (4.99) be0
low, so that our task is only to verify the assumption (4.7) modied for a ve torvalued perturbation parameter
" # ("1 ; "2 ).
" ¡ 0 and y0" are bounded thanks to (4.81d), we an see by the Gronwall inequality that the olle tion { y " } " ¡0 is bounded n 1 q n in L ( I ; R ), where y " ò W ( I ; R ) denotes the unique solution to the initialvalue First, as (4.70g) holds uniformly with respe t to
;
problem
dy " # f " (t ; y " ; u) ; dt
By (4.81a) and (3.192 ) (used for
f"
y " (0) # y0" :
(4.99)
and modied in the spirit of Remark 3.104), we
an further estimate the dieren e between (4.99) and the unperturbed equation as follows
!! ! !! y " ( t )" y ( t )!!!
!! t
# !!!!X f " ( ; y " ; u) " f( ; y; u) d % y " " y !
¢
0
0
t " X f ( ; y " ; u ) " f ( ; y " ; u )d 0
0
t
!! !! !! !
% X f( ; y " ; u) " f( ; y; u)d % y " " y 0
0
0
t
¢ X (a () % b (y " ()) % u()p )"d 0
0
%
0
t X a2 () 0
from whi h we obtain
0
% b (y " ()) % b (y()) % u()p y " ()"y() d % " ; 2
2
2
lim"ÿ y " " y C I Rn # 0 by the Gronwall inequality. 0
( ;
)
Ë 293
4.3 Optimal ontrol of nitedimensional dynami al systems
By (4.81 ) we then get
"
y" "
y C I R ¢ " y " " y " C I R % y " " y C I R ¢ " % o % ( y " " y L ( ;
( ;
)
( ;
y
)
I
Rn ) ) ;
( ;
is the modulus of ontinuity53 of on I , { r ¢ % } where % is so large that ¢ % and y " C I Rn ¢ %. This veries (4.7a) for R " (u) # " y " with y " # " (u)
o%
where
)
Rn )
C(I;
( ;
)
solving (4.99). By (4.81b) and (3.192 ) used for
"1
;
" 2 ( u ) " ( u )
¢
' " , one an further estimate
T ! " ! X !!! ' 1 ( t ; y " 2 ; u ) " ' ( t ; y " 2 ; u )!!! d t 0 T %X !!!!'(t ; y "2 ; u) " '(t ; y; u)!!!! dt 0
¢ a % b (y "2 ) % up L1 I " % a % b (y "2 ) % b (y) % up L1 I y "2 " y C I Rn : 0
0
2
Thus (4.7b) modied for
0
( )
2
2
1
2
( ;
( )
)
" # ("1 ; "2 ) has been veried.
Å
As for (4.7 ), it follows simply from (4.70h) ombined with (4.81b).
Remark 4.48 (Delayed ontrols).54 The ontrol onstraint of the type (4.63b) need not be always satisfa tory. E.g., one an onsider a problem with one additional delayed
ontrol (
t0 ¡ 0 is a xed time delay):
Minimize
T X '(t ; y(t); u(t); u(t 0
subje t to
"t
0 ))
dt
dy # f(t ; y(t); u(t); u(t " t )) for t ò I; y(0) # y ; dt u(t) ò S (:a.a. t ò ("t ; T)); y ò W q (I; Rn ); u ò L p ("t I ; Rm0 ): 0
0
0
0
1;
/ 7 7 7 7 7 7 7 ? 7 7 7 7 7 7 7
(4.100)
G
m # 2m0 , S(t) # S0 , S0 , and u $ (u1 ; u2 ) ò L p (I; Rm0 )2 , namely
ODE It an obviously be transformed into the form (POC ) with
one additional onstraint on the new ontrol
u2 (t) # u1 (t " t0 )
:a.a. t ò (t ; T) : 0
ODE The relaxed problem takes again the form (R POC ) but the expli it form of
H
(4.87) is now quite deli ate matter. For the spe ial ase
S0
U ad
from
Rm0 ompa t and H #
ò C(I , Rn ; R ), is uniformly ontinuous on ea h ompa t I , {r ¢ %}. In parti ular, there is o % : R Ù R% su h that lim"ÿ o % (") # 0 and (t ; r ) " (t ; r ) ¢ o % (max(t " t ; r " r )) whenever max(r ; r ) ¢ %. 53
2
54
Let us note that, sin e
%
1
2
0
1
1
1
2
2
1
2
Delayed ontrols has been treated by Rosenblueth [651653℄ and Vinter [654℄, and by Warga and
Zhu [794, 815℄.
294
Ë
4 Relaxation in Optimization Theory
L1 (I; C(S0 , S0 )), the onvex set of admissible relaxed ontrols U ad Y(I; S0 , S0 ) was expli itly des ribed in Rosenblueth's works [651653℄ as
U ad # ò Y(I; S0 , S0 ); :h ò L1 (t I ; C(S0 )) : T X X h(t ; s2 ) t (ds1 ds2 ) dt t0 S0 ,S0
T
#X
X h(t ; s1 ) t"t0 (ds1 ds2 ) dt : t0 S0 ,S0
Remark 4.49 (Relaxation via a onvex ompa ti ation by J.E. Rubio).
A
ompletely
different approa h is an attempt to ompa tify the pairs of ontrolstate whi h
dy dt
# f(t ; y(t); u(t)) on I with y(0) # y0 . Let us H Car(I , (Rn ,S0 )) with S0 Rm ompa t and the embedding i H : ( y; u ) ÜÙ H . For ò C 1 (I , Rn ), when putting
satises the state equation, i.e. here present it briey by onsidering *
f
(
(
t ; r; s) #
)
%
t ; y(t)) dt #
(
r ( t ; r ) f ( t ; r; s )
t (t ; r) ;
we note that
T X 0
f
T
(
t ; y(t); u(t)) dt # X
0
d dt
(
T; y(T)) "
0; y ) :
(
0
$ 0) and with $ 0 and
ODE Considering (POC ) without the state onstraints, (i.e.
S(t) # S0 , based on [778℄ one an think about its metamorphosis into Minimize subje t to
i H ( y; u ) ; '
# (T; y(T)) " (0; y0 ) : ò C u(t) ò S0 (:a.a. t ò I ) ; 1 q n m ( y; u ) ò W (I; R ) , L (I; R ): f
i H ( y; u ) ;
;
(1)
(
I ,R
/ 7 7 7 n) ; 7
? 7 7 7 7
(4.101)
G
and then the relaxed problem takes the form: Minimize
; '
subje t to
;
T; y(T)) " òr a% (I , Rn , S0 ) : f
#
(
0; y
(
0)
: òC
(1)
(
I , Rn ) ;
/ 7 ? 7 G
(4.102)
In a series of works started by [707, 708℄ and summarized in the monograph [709℄, it is shown that the measures admissible in (4.102) are weakly* attainable by sequen es of the pairs of the original state ontrols.55 The set of these measures is obviously
r a(I , Rn , S ) and thus it forms a onvex  ompa ti ation of q ( I ; R n ) , L ( I ; R m ); d y # f ( y; u ) ; y (0) # y }. Optimality onthe set {( y; u ) ò W dt
onvex subset of
0
1;
0
ditions based on the geometry of this onvex ompa ti ation was then formulated in [777, Thm 2.2℄. An approximation by dis retising the set of measures
55
r a% (I , Rn , S
y(t) # y T xed, and then it is proved that any for (4.102) an be attained by an admissible sequen e for (PODE OC ) but with this terminal
A tually, [709℄ onsiders the terminalstate onstraint
admissible
0)
state onstraint satised only asymptoti ally in the spirit used already in Proposition 4.1.
4.3 Optimal ontrol of nitedimensional dynami al systems
Ë 295
(i.e. an inner approximation) or by taking only a nite number of onstraints by taking nite number of test fun tions
's (i.e. an outer approximation) would lead to
a semiinnite mathemati al programming (SIP) be ause either the number of linear
onstraints or the number of variables still remain innite; here it is semiinnite linear programming. The outer approximation may underrelax the problem but in the limit when number of onstraints in reases onverges to the orre t relaxation. Combining both approximation then leads to a linear mathemati al programme (LP) that
an be e iently implemented on omputers even for a relatively very large number of variables and onstraints, as the approximate problems presumably have. The Rubio's onstru tion has later been reinvented under the name linearmatrixinequality (LMI) relaxations and the measures o
urring in (4.102) alled o
upation measures, f. e.g. [29, 219, 220, 478, 479, 511℄, together with a ombination of the method of moments for numeri al approximation when the nonlinearities are polynomial as in Se t. 3.3.d.
4.3.
Optimality onditions
The further aim of ours is to exploit the results from Se tions 4.1 and 4.2 to ompose ODE the optimality onditions for (R POC ). Of ourse, we must strengthen (4.88) and (4.91)
H
to
f òCAR H di (I , Rn , Rm ; Rn ) ;
p; q
and
;
' òCAR H di (I , Rn , Rm ; R): ;
p;1
;
(4.103)
Proposition 4.50 (Maximum prin iple). Let H be a C(I)invariant subspa e of Carp (I; Rm ), p ò [1; %), q ò (1; %), the one D R has a nonempty interior, n k , n ) with r ò C(I , R ; R r ( t ; r ) # ( t ; r )/ r , (4.70g), (4.70h) and (4.103) be valid, ODE and let ( ; y ) ò Argmin(R H P ). Then there are £ 0 and ò r a(I; R ) su h that ( ; ) #Ö 0, £ 0, the omplementarity ondition (4.72) is valid, and the integral
*
*
*
*
0
OC
*
*
0
maximum prin iple
T X [ h y; 0 ; *
0
*
Ǳ ℄(dt) # sup
*
u ò U ad
is valid, where the Hamiltonian
h y 0 ;
*
;
*
T X h y; 0 ; 0
(
*
t ; u(t)) dt
(4.104)
ò H is given by (4.71)56 with ò L q (I; Rn ) solving
*
the ba kward terminalvalue problem:57
d % f r y Ǳ # ' r y Ǳ % dt *
*
*
0
56
In fa t, we derive uniquely
# 0.
h y;0 ; *
*
only up to
r
y ; (T) # r (y(T)): *
*
*
0
1òH
for
(4.105)
ò L1 (I) arbitrary. We hoose simply
* Of ourse, (4.105) is to be understood in the sense of distributions. In fa t, always belongs to 1/ " ; p; 1 L (I; n ). Moreover, if # 0 and if ' ò CARH;di (I , n , m ; ) for some " ¡ 0, then even * ò W 1;min(q;1/(1"")) (I; n ) be ause 'r y Ǳ ò L1/(1"") (I; n ) thanks to the growth ondition (3.197b). In
57
R
R
R
R R R
this latter ase, the solution to (4.105) an be understood in the usual Carathéodory sense.
Ë
296
4 Relaxation in Optimization Theory
S is measurable losedvalued, and if S(t) is bounded in Rm uniformly with respe t to t or 0 ¡ 0, then h y Ǳ is absolutely ontinuous and the following point0 1 wise maximum prin iple is valid in the sense of L ( I ): Moreover, if
*
;
h y 0
*
;
;
*
*
*
;
Ǳ (t) # sup h y 0 ;
sòS(t)
*
;
(
*
:
t ; s))
a.a.
tòI :
(4.106)
Proof. We will use Theorem 4.15 to the problem (RPOC ) with the data from (4.63) and with
F
#
FH
from (4.29). Su h problem transformed via
*
: FH Ù H *
*
is equiv
ODE alent to (R POC ). The smoothness assumptions of Theorem 4.15 are guaranteed via H
was D has a nonempty interior in C(I; R ),
Lemma 3.103 while the ontinuity58 of the extended ontroltostate mapping proved in Lemma 4.45. Also note that the one
int( D ) #Ö .
as required in Theorem 4.15 be ause
For larity, we divide the derivation of the above laimed optimality onditions into separate steps.
(; y), B (; y), and J (; y).) In view of Lemma 3.103 the parq n tial dierentials ( ; y ) ò L( H ; L ( I ; R ) , R), B ( ; y ) ò L( H ; C ( I ; R )), and J ( ; y ) ò L( H ; R) are given respe tively by the formulae: Step 1. (Dierentials of
*
*
*
with
òH
*
[
( ; y)℄( ) # "f y Ǳ ; 0 ;
[
B ( ; y)℄( ) # 0 ;
[
J ( ; y)℄( ) #
T X ['
0
y Ǳ ℄(dt) # ; ' y
. Moreover, all the dierentials are (weak*,weak) ontinuous59 as required
by Theorem 4.15.
Step 2. (Dierentials
of
the partial dierentials
L( W
1;
q (I;
R
n ); C(I;
R
)),
( ; ), B ( ; ), and J ( ; ).) In view of Lemma 3.103 1 q n q n n y ( ; y ) ò L( W ( I ; R ) ; L ( I ; R ) , R ), y B ( ; y ) ò 1 q n and y J ( ; y ) ò L( W ( I ; R ) ; R) are given respe tively by
;
;
the formulae:
dy " (f r y Ǳ )  y ; y (0) ; dt [ y B ( ; y )℄( y ) # r y
 y ;
[ y
( ; y)℄( y ) #
J ; y)℄( y ) #
[ y (
58
T X [' r 0
y Ǳ ℄(t)  y (t) dt % (y(T))  y (T)
R
Y # W 1; q ( I ; n ) is (strong,strong) ontinuous, as re
In fa t, the linear and the nonlinear parts should be treated separately: we endow
by the norm of
L (I;
Rn
) so that the ontroltostate mapping
quired by Theorem 4.15 be ause of Lemma 1.59. Let us note that the respe tive dierential of the nonlinear part remains ontinuous with respe t to this weaker norm, while the dierential of the linear part, being onstant, an be treated in the original strong topology of
59
As for [
( ; y)℄, this requires g
 (
f
W 1; q ( I ;
Rn
).
y) ò H for any g ò L q (I; Rn ) Ê L q (I; Rn )
*
, whi h is just
J ; y)℄, it follows simply from ' y ò H .
ensured by (3.196) whi h is ee tive due to (3.46). As to [ (
4.3 Optimal ontrol of nitedimensional dynami al systems
y ò W 1 q (I; Rn ). ;
with
( ; y)
Let us note that y
Ë 297
a tually possesses a bounded in
verse,60 as required in Theorem 4.15.
Step 3. (The adjoint problem.) The abstra t adjoint equation (4.23b) bears the form
´
; [y ( ; y)℄( y )µ # 0 ´y J ( ; y); y µ % *
*
´
; [y B ( ; y)℄( y )µ
*
(4.107)
y ò W 1 q (I; Rn ) and some 0 £ 0, $ (1 ; 2 ) ò L q (I; Rn ) , Rn , and £ 0 su h that ( 0 ; ) #Ö 0, < ; B ( y; )> # 0; note that the last identity just results to the *
;
for all
*
*
*
*
*
*
*
*
omplementarity ondition (4.72). Using the formulae from Step 2, the identity (4.107) takes the form
T * X 1 0
dy "  f r y Ǳ  y dt %  y (0) dt T ' r y Ǳ (t)  y (t) dt % (y(T))  y (T) %X (
*
1
T X
#
*

*
*
0
0
2
0
y)  y  (dt) :
*
r
0
(4.108)
Using the bypart integration, we an easily see that (4.108) will be valid for every
y ò W 1 q (I; Rn ) provided # (1 ; 2 ) ò L q (I; Rn ) , Rn ;
*
*
*
butions):
satises (in the sense of distri
d %  f r y Ǳ # " ' r y Ǳ " dt (T) # " (y(T)) and (0) # : *
*
1
1
*
*
0
*
*
1
0
*
*
1
2
 (
r
instead of
"
(4.109a) (4.109b)
From the last equality, we an eliminate the formal multiplier *
y) ; 2 , and *
write simply
*
. Thus we ome just to (4.105). Let us note that, thanks to (3.197b), 1
the righthand side of the linear ordinary dierential equation in question, namely
0 ' r y Ǳ % r y) , belongs to r a(I; Rn ) whi h is ontained61 in W 1 q (I; Rn ) . As y has a bounded inverse as shown in Step 2, our terminalvalue problem possesses q n always a (unique) solution ò L ( I ; R ), as required. *
*
;
*
*
Step 4. (The Hamiltonian.) The abstra t Hamiltonian (4.23d) an be now written in the form
f y 0 ;
*
;
*
# h y 0 ;
*
;
*
with
h y 0
*
;
;
*
ò H determined with help of the formulae from
Step 1 by the identity
; h y; 0 ; *
*
# ; [ ( ; y)℄( ) " ; [ B ( ; y)℄( ) " *
*
# ; " f y Ǳ " ; ' y # ; " *
*
1
whi h is to hold for any
h y 0 ;
*
;
òH
*
0
*
1
 (
J ( ; y ) ;
0
f y) " 0 ' y ; *
. This gives the expression (4.71) for the Hamiltonian
if we write shortly, as in Step 3, *
p; q
*
*
in pla e of
R R R
"
R
*
1
. As
f y 0 ;
*
;
is determined *
f ò CARH;di (I , n , m ; n ) ensures f r y Ǳ ò L q (I; n,n ), and therefore the initialvalue problem d y/dt " (f r y Ǳ )  y # f and y(0) # y0 denes the bounded linear operator (f; y0 ) ÜÙ y : L q (I; n ) , n Ù W 1;q (I; n ) being just [y ( ; y)℄"1 . 1; q n ) C ( I ; n ) is of the type (D) and therefore the 61 Here we use the fa t that the embedding W (I; 60
Indeed,
R
R
;
R
R
R
adjoint operator realizes the ontinuous embedding; f. Se tion 1.3 .
Ë
298
4 Relaxation in Optimization Theory
ò H essentially does not hange by T adding arbitrary integrand of the form 1 ò H be ause ( 1) # onst.# P ( t ) d t . h y 0
uniquely up to onstants, our Hamiltonian
;
*
;
*
0
Step 5. (Lo alization of the maximum prin iple.) Eventually, the maximum prin iple (4.23 ) an be transformed into the form (4.106) by means of Theorem 4.21(i). Let us verify the des ent ondition (4.36). It is satised trivially when all
S(t) are
t
bounded independently of . Also, (4.70g) and (4.70h) allow us to estimate
h y 0
*
;
;
(
*
t ; s) ¢ "0 a(t) " 0 bsp % (t) a1 (t) % 1 sp *
*
*
/
q
% y C I Rn ;
1
( ;
)
0 ¡ 0 be ause then one an estimate (t) 1 sp q (1 % 1 p y L I ;Rn ) ¢ 2 0 b s % C with C large enough depending on 0 , 1 , on y L I ;Rn , and on L I ;Rn ; re all that q ¡ 1. Having (4.36) at our disposal, we an readily use *
whi h gives (4.36) provided
*
*
(
*
)
*
(
/
(
)
Å
Theorem 4.21(i) to get (4.106).
Remark 4.51 (Setting the state equation alternatively). dene : U , Y Ù X as Y # L q (I; Rn ); X # W 1 q (I; Rn ) ; ;
(u ; y);
T
y # X y
0
)
*
and, for all
Instead of (4.63a, ,e), one an
y ò W 1 q (I; Rn ) : ;
dy % f(y; u)  y dt " y(T)  y (T) % y dt
y (0) :
This integral identity indeed overs both the state equation
0

d dt y
(4.110)
# f(y; u) on I
and
y(0) # y0 and, under the same data quali ation as before, the y ò L q (I; Rn ) to the state equation (u ; y) # 0 does exists, even belongs to W 1 q (I; Rn ) as before, and is unique for a given u.62 Then, instead of the al ulations
the initial ondition solution ;
(4.108)(4.109), the abstra t adjoint equation (4.107) results to
d %  f r y Ǳ  y dt " (T)  y (T) dt T # X ' r y Ǳ (t)  y (t) dt % (y(T))  y (T) %
T X
*
y
*
*
0
*
*
0
0
0
T X ( 0
r
y)  y  (dt) : *
(4.111)
y ò W 1 q (I; Rn ). From this, we an read that ò X # W 1 q (I; Rn ) satises the n terminalvalue problem (4.105) in the sense of R valued measures on I . for any
;
*
*
;
Corollary 4.52 (Chattering ontrols I). be fullled,
H
be separable,
Let all the assumptions made in Proposition 4.50
S be measurable and losedvalued, and, for a.a. t ò I , any
# f(u ; y) a.e. on I an be seen by taking an arbitrary y ò C (I; Rn ) with y(0) # 0 # y(T) and making the bypart integration in time. This reveals also that y ò W q (I; Rn ) and, after bypart integration in time, (4.110) results to ( y (0)" y )  y(0) # 0. Taking now y arbitrarily gives also the initial ondition y (0) # y . Having two solutions to ( u ; y ) # 0 and ( u ; y ) # 0, we an use that 62
The equation
d dt y
1
1;
0
they satises also as before.
d dt y 1
0
# f(u ; y
1 ) and
d dt y 2
# f(u ; y
1
2
2 ) and show uniqueness by Gronwall's inequality
4.3 Optimal ontrol of nitedimensional dynami al systems
Ë 299
0 ¡ 0 and any r; r ò Rn , the fun tion "0 '(t ; r; ) % r  f(t ; r; ) attains its maximum on S(t) at no more than k points. Then every optimal ontrol for (RPODE ), whi h admits the
orresponding multiplier 0 positive, is k atomi . *
*
*
*
OC
*
must satisfy the maximum prin ih # h y 0 whi h satisfy
Proof. By Proposition 4.50, every optimal ontrol
ple (4.106). Then it su es to apply Proposition 4.27(i) with
;
*
;
*
0 ¡ 0, f. Step 5 of the proof of Proposition 4.50.
Å
*
the des ent ondition (4.36) if
Corollary 4.53 (Chattering ontrols II).63
Let all the assumptions made in Proposi
tion 4.50 be fullled, H be separable,
S be measurable and losedvalued, and there is at least one optimal solution for whi h 0 ¡ 0. Then there exists at least one ( n %1)atomi *
hattering optimal ontrol.
0 , whi h does exist due to Proposi0 and
Proof. Let us take some optimal relaxed ontrol tion 4.46, for whi h
0 ¡ 0. Furthermore, let *
*
be the adjoint state related with
h y 0 be the Hamiltonian from (4.71). Then it su es to apply Proposition 4.28 with h # h y 0 ò H and h l # [f y℄l for l # 1; :::; n. Note that h satises the des ent ondi;
*
;
*
;
*
;
*
tion (4.36), f. Step 5 of the pre eding proof. Let us verify the ondition (4.49), onsider
ò U ad su h that h Ǳ # h S with h S (t) :# supsòS t h(t ; s) and h l Ǳ # h l Ǳ 0 . First, let us noti e that the last ondition ensures that ( ) # ( 0 ); in other words, both and 0 drive the ontrolled system to the same state y . As we supposed 0 ¡ 0 we an write ing some other relaxed ontrol
( )
*
' yǱ #
# whi h implies
1 0
*
1
0
*
*
*
 (
f y Ǳ ) " h y 0
 (
f y Ǳ 0 ) " h y 0
;
*
;
;
*
*
;
J ( ; y) # J (0 ; y) # min(RH PODE OC ).
Ǳ %
*
Ǳ 0 %
# (' y) Ǳ 0 ;
Therefore
is optimal for (R
ODE H POC ),
verifying thus the hypothesis (4.49). Then our assertion follows from Proposition 4.28 with
k # n.
Å
We would like to point out that the estimates annot be improved in the sense that one an onstru t examples that do not admit any hattering relaxed ontrol with less atoms than stated in Corollaries 4.524.53; f. also the example in Subse tion 4.3.e. In some ases where the ontrolled system is only slightly nonlinear in terms of the states, the relaxed problem an be proved onvex. Then the rstorder optimality
63
Su h kind of results (but a more pessimisti (
n%2)atomi estimate)
was outlined also by Cesari
'(t ; r; s) $ '(t ; r)), then the existen e of an (n%1)atomi ontrol was established too; f. [196, Se t. 1.14A℄. Supm ompa tvalued, in [196℄ su h results were derived dire tly from Proposition 4.50 posing S : I Â± by means of the te hnique of the proof of Proposition 4.28. The ( n %2)atomi ontrols have been also [196, Se t. 1.14B℄. If the ost fun tional would not depend expli itly on the ontrol (i.e.
R
used by Berkowitz [110, Chap. IV℄ and Carlson [170℄.
300
Ë
4 Relaxation in Optimization Theory
ondition (i.e. the maximum prin iple) is not only ne essary but also su ient for optimality. Let us illustrate it on the additively oupled ase, whi h allows to opy the arguments behind the abstra t Proposition 1.62 in the on rete situation.
Proposition 4.54 (Su ien y of the maximum prin iple).64 For the additive ansatz '(t ; r; s) # g(t ; r) % h(t ; s); f(t ; r; s) # G(t ; r) % H(t ; s); let us assume that
# 0;
and
(4.112)
g : I , Rn Ù R, G : I , Rn Ù Rn , h : I , Rm Ù R, H : I , Rm Ù Rn
are Carathéodory fun tions satisfying the growth onditions
;a ò L q (I) ;b ò R : ;a ò L (I) :
1
with some
G(t ; r) ¢ a(t) % br;
g(t ; r) ¢ a(t);
H(t ; s) ¢ a(t);
(4.113a)
h(t ; s) ¢ a(t)
(4.113b)
q ò (1; %), and a smoothness onditions
;a ò L (I) ;b : R Ù R ontinuous : g (t ; r) ¢ a(t) % b(r); g ( t ; r ) " g ( t ; r ) ¢ ( a ( t ) % b ( r ) % b ( r )) r " r ; % " ;a ò L (I) ;b : R Ù R ontinuous : G (t ; r) ¢ a(t) % b(r); G ( t ; r ) " G ( t ; r ) ¢ ( a ( t ) % b ( r ) % b ( r )) r " r : 1
1
2
1
1
2
1
2
1
2
1
2
1
2
(4.113 ) (4.113d) (4.113e)
S : I Â± Rm is supposed bounded, measurable, and in the m m m form S ( t ) # M ( t ; S 0 ) for some S 0 R ompa t and M : I , R Ù R a Carathéodory " 1 mapping su h that both M ( t ; ) and M ( t ; ) are Lips hitz ontinuous uniformly with respe t to t ò I . Let us assume G ( t ; ) twi e ontinuously dierentiable, and let g ( t ; ) be The multivalued mapping
uniformly onvex in the sense
:r; r ò Rn : max(r; r ) ¢ R âá
g(t ; r ) " g(t ; r) " g (t ; r)( r " r) £ (t) r "r2
with
(i)
(4.114a)
R[y(u)℄(t) and with the modulus £ 0 satisfying
(t) £ with
b(t)
2
eB t sup G (t ; r) ( )
r ¢R
with
b(t) :#
T X a ( ) d and t
B(t) :#
T X A() d t
(4.114b)
a(t) :# sup r ¢R g (t ; r) and A(t) :# sup r ¢R G (t ; r).65 Then: is onvex on U ad , and
(ii) the maximum prin iple (4.105)(4.105) with some
0 ¡ 0 and £ 0 *
ient ondition for the relaxed ontrol to be optimal.
64
For a generalization for unbounded ontrols see [675℄.
65
Note that (4.113 .d) ensures
a ; A ò L1 (I).
*
*
is also a suf
4.3 Optimal ontrol of nitedimensional dynami al systems
Ë 301
Proof. The maximum prin iple involves the adjoint equation
d # " (t)G (t ; y(t)) " g (t ; y(t)) ; (T) # 0 : dt *
*
*
(4.115)
The assumption (2.2) ensure that the terminalvalue problem (2.3) possesses pre isely
ò W 1 1 (I; Rn ). ; ò U ad and y; y ò W 1 q (I; Rn ) solve the initialvalue problem in (RP) with
one solution
;
;
Let and
*
, respe tively. Then using the bypart integration and the adjoint equation (4.115),
we an al ulate:
(
)
" ( ) " [(
)℄(
" (
T X X
(
)
T
#X
"
)
0
"
Rm
)
*
t H(t ; s) % h(t ; s)[
( )
y (t)) " g(t ; y(t)) " X
g(t ;
(t)H(t ; s)[
t
"
t ℄(d s ) d t
t
"
t ℄(d s ) d t
*
Rm
0
d( y (t) " y(t)) dt dt T d # X g(t ; y (t)) " g(t ; y(t)) % (t)(G(t ; y (t)) " G(t ; y(t))) % ( y (t) " y(t))dt dt T
# X g(t ; y (t)) " g(t ; y(t)) % (t) G(t ; y (t)) " G(t ; y(t)) " *
0
*
*
0
T
# X g(t ; y (t)) " g(t ; y(t)) " g (t ; y(t))( y (t) " y(t))
#: g ( t )
0
% (t) ( G ( t ; y ( t )) " G ( t ; y ( t )) " G ( t ; y ( t ))( y ( t ) " y ( t ))) dt *
(4.116)
#: G ( t )
g (t)
Estimating the se ondorder orre ting terms
G (t),
and
the in remental
formula (4.116) enables us to investigate onvexity of the extended ost fun tional From the adjoint equation (4.115) we an estimate
d * dt
.
¢ A(t) (t) % a(t) so that by *
the Gronwall inequality one gets
(t) ¢ *
T " P T A()d d ePtT A()d X a ( )e t t
: (t) ¢ " G(t ; y(t)) "
To simplify the notation, we an also (a bit more pessimisti ally) estimate
*
b(t)eB t : By the Taylor expansion, we an estimate G(t ; y (t)) G (t ; y(t))( y (t)" y(t)) ¢ sup r ¢R 12 G (t ; r) y (t)" y(t)2 . Then (4.114) ensures ( )
1 ( t ) G ( t ; y ( t )) y ( t ) " y ( t ) 2 1 sup G (t ; r) y (t) " y(t) £ 0 : r ¢R 2
g (t) % (t) G (t) £ (t) y (t) " y(t)2 " *
£ (t) "
b(t)
2
eB t
( )
*
2
2
so that the se ond righthand term in (4.116) is nonnegative. From (4.116) we obtain
: ; ò U ad :
whi h just says that
ondition then follows.
(
is onvex on
)
" (
U ad .
)
" [(
)℄(
"
)
£ 0;
The su ien y of the 1storder optimality
302
Ë
4 Relaxation in Optimization Theory
Example 4.55 (Conventional relaxed ontrols).66Let us apply our theory to the problem ODE m (POC ) with S ( t ) losed and bounded uniformly in time, i.e. S ( t ) S for a ball S in R . 0
Then the growth we keep
pò
0
p of the ontrols is irrelevant; nevertheless, for notational simpli ity,
R% formally in our problem. The most general s heme will be reated 0
by taking the nest possible relaxation from the onsidered lass, reated obviously by
H # Carp (I; Rm ). We may and will endow this H
by the universal (semi)norm ( f.
(3.141)):
h H #
Then the restri tion operator
inf
:(t ; s)òI ,Rm: p h ( t ; s )¢ a ( t )% b s
a L1 I % b :
(4.117)
( )
h ÜÙ hI ,S0 : Carp (I; Rm ) Ù L1 (I; C(S0 ))
is linear and
ontinuous.67 Besides, this restri tion mapping is also surje tive so that the adjoint
L1 (I; C(S0 )) Ê Lw (I; r a(S0 )) Ù H # Carp (I; Rm ) is ontinuous and inje tive, and embeds the set of Young measures Y( I ; S 0 ) L w ( I ; r a( S 0 )) ( f. also Examp m m ple 3.44) into Y ( I ; R ). Thus U H ad is anely homeomorphi with { ò Y(I; R ); :a.a. t ò I : supp( t ) S(t)} provided S satises some additional quali ation, e.g. (3.30). *
operator
*
*
*
*
ODE Then (R POC ) an be equivalently written in terms of the lassi al relaxed ontrol
H
(= Young measures) as follows
T X X
Minimize
0
Rm
'(t ; y(t); s)
t (d s ) d t % ( y ( T ))
/ 7 7 7 7 7 7 7 7 7
dy # X f(t ; y(t); s) t (ds) (:a.a. t ò I); y(0) # y ; dt Rm ? 7 7 ( t ; y ( t )) ¢ 0 (: t ò I ) ; 7 7 7 7 7 supp( t ) S ( t ) (:a.a. t ò I ) ; 7 7 m q n y ò W (I; R ); ò Y(I; R ): G
subje t to
0
(4.118)
1;
Therefore, under the respe tive data quali ation, we are authorized to apply Propositions 4.46 and 4.50 to this on rete problem; note that the oer ivity ondition (4.70h) is fullled automati ally be ause
S(t)
are here bounded uniformly for
t ò I.
In par
ti ular, by Proposition 4.46 we have guaranteed the existen e of an optimal relaxed
ontrol
and, by Proposition 4.50, we have at our disposal the pointwise maximum
prin iple (4.106), whi h an be now written in the form
h y; 0 ; X S(t) *
66
(
*
t ; s)
t (d s )
# max h y 0 (t ; s) sòS(t)
;
*
;
(4.119)
*
Su h kind of relaxation was frequently used in the literature; let us mention for example Balder
[5052℄, Barron and Jensen [79℄, Berkowitz [111℄, Carlson [170℄, Gamkrelidze [345℄, GhouilaHouri [352℄, Goh and Teo [358, 758℄, M Shane [528℄, Medhin [531℄, Papageorgiou and Papalini [592℄, Rishel [644℄, S hwarzkopf [724℄, Warga [786791℄, Williamson and Polak [798℄, et .
67
there are
a " ò L (I) and b " ò 1
"
"
" " 1 " " h I ,S0 " "L (I;C(S0 )) ¢ max(1 ; T maxsòS0 s ) h H be ause for any " ¡ 0 % su h that a 1 % b # h % " and h ( t ; s ) ¢ a ( t ) % b s p , and then
Indeed, we have the estimate
R
p
" L (I) " T P a " (t) 0
# PT maxsòS0 h(t ; s) dt ¢ ¢ max(1; T maxsòS0 sp )( h H % "). " " " " 1 " " h I ,S0 " "L (I;C(S0 ))
0
H
"
"
% b " maxsòS0 sp dt ¢ a " L1 I % Tb " maxsòS0 sp ( )
Ë 303
4.3 Optimal ontrol of nitedimensional dynami al systems
for a.a.
tòI
h y 0
with the Hamiltonian
*
;
;
dened by (4.71) with *
*
solving the adjoint
equation problem68
d % X f r (t ; y(t); s) (t) t (ds) dt S t # X ' r (t ; y(t); s) t (ds) % *
*
( )
*
0
S(t)
Example 4.56 (A universal approa h).
(
t ; y(t)) ;
y(T) # 0 (y(T)) :
*
*
Let us investigate a general situation when
S(t) need not be bounded uniformly with respe t to t ò I . Supposing (4.70h), we have L p (I; Rm ) but not L (I; Rm ), and therefore we are for ed to
got the oer ivity only in
employ the general theory from Se tion 3.4. To extend as mu h problems as possible, we shall ertainly take the nest onvex ompa ti ation from the investigated lass. This means we put here
H # Carp (I; Rm ) ;
(4.120)
endowed with the universal (semi)norm (4.117). Then, in fa t, only the natural growth
onditions, i.e. (4.70af), are imposed on the Carathéodory integrands
f and ' be ause
the assumptions (3.192a), (3.196), and (3.197a) are void, as shown in Remark 3.107. However,
H
from (4.120) is not separable, whi h eventually prevents any usage of a great
part of our results. This is the reason why smaller subspa es ations, are more advantageous. In fa t,
H , reating oarser relax
H should only ontain all possible integrands
that an appear in the investigated problem(s). Having in mind only a single problem with the data
f
and
' satisfying (4.70af), one an put
H # span g  (f y) % g  (f r y)  y % g0  (' y) % g 0  (' r y)  y ;
y; y ò C(I; R ); g0 ; g 0 ò C(I); g; g ò L n
q
(
I; R
n
)
:
(4.121)
C(I)invariant linear subspa e of Carp (I; Rm ) and also (4.103) is satised. Moreover, H from (4.121) is separable69 if endowed with the norm Let us note that su h
H
is a
(4.117). If we are not interested in optimality onditions, we an avoid the data quali ation (4.70 ) and (4.70f) and take a smaller
H , namely
H # span g  (f y) % g0  (' y); y ò C(I; Rn ); g0 ò C(I); g ò L q (I; Rn ) ;
whi h is a separable
C(I)invariant
linear subspa e of
(4.122)
Carp (I; Rm ). Then (4.88) and
ODE (4.91) are guaranteed. Moreover, dealing with a olle tion of the problems (POC ; " ; " ), 1 2
we an take a linear hull of all subspa es onstru ted in (4.121) or (4.122). If taking the
olle tion ountable, we do not lose the separability of the resulted subspa e
68
H.
The solution of the adjoint problem is to be understood in the distributional sense sin e
*
is in
general a measure.
C and L q spa es involved in (4.121) and from the separate (strong,strong) ontinuity of the mappings ( g; y ) ÜÙ g  ( f y ), ( g; y; y) ÜÙ g  (f r y)  y, (g; y) ÜÙ g  (' y), and ( g; y; y) ÜÙ g  ('r y)  y; for the ontinuity with respe t to the yvariable we refer to Remark 3.107 while the ontinuity with respe t to g  and yvariables is an easy exer ise; f. also Proposition 3.102. 69
This follows from the separability of the
304
Ë
4 Relaxation in Optimization Theory
Example 4.57 (Linear/ onvex problems).
A
great deal of problems appearing in
appli ations have got a linear/ onvex stru ture (
t ; r; ) D  onvex,
and
S
'(t ; r; )
onvex,
f(t ; r; )
ane, and
onvexvalued, measurable losedvalued. If our growth
assumptions as well as the oer ivity assumption (4.70h) with
p ¡ 1
are fullled,
then su h problems do not require any relaxation at all. In fa t, it su es to endow
L p (I; Rm ) by the weak topology (re all that always p %). In other words, we an p m make the onvex  ompa ti ation of the original spa e of ontrols L ( I ; R ) by means of the subspa e
H # L p (I) (Rm ) ;
whi h auses
*
(4.123)
YH (I; Rm ) Ê L p (I; Rm ) and thus U ad Ê Uad ; f. also Examples 3.50 and p
3.73. It is well known70 that the ost fun tional in the original problem is weakly lower ODE semi ontinuous, so that the original problem (POC ) essentially oin ides with the reODE laxed problem (R H POC ) if one admits a nonane extension of the ost fun tional, i.e.
the term (
' y) Ǳ is repla ed by N 'y (). In parti ular, by Proposition 4.2(i) there is
no relaxation gap. On the other hand, if the data satisfy the mild assumptions (4.70a)(4.70g) with
p # q ¡ 1, then one an make also a ner relaxation by taking H , i u
ontrol also for the nely relaxed problem whenever u is the optimal ontrol for the the natural hoi e
e.g., as in (4.121). We an apply here Corollary 4.36 be ause H ( ) is an optimal relaxed
oarsely relaxed (i.e. here original) problem.71 This yields the lassi al maximum prin
h y 0 òCarp (I; Rm ) resulting from the ner relaxation svariable, h y 0 : Uad Ù R is weakly upper semi ontinu
iple (4.75). As the Hamiltonian is here on ave in the
;
*
;
*
*
;
*
;
ous,72 and therefore we an use Proposition 4.10 to transfer this maximum prin iple on the oarser relaxation, whi h gives again the maximum prin iple (4.75). If additionally
'(t ; r; ) is dierentiable, then h y 0 ;
*
;
t ; ) is smooth and on ave, so that we an h ò N S t (u(t)) or, more expli itly
(
*
rewrite the ondition (4.75) into the form ( y; * ; *) s 0
( )
f s (t ; y(t); u(t)) (t) " 0 ' s (t ; y(t); u(t)) ò N S
:a.a. t ò I :
*
*
t
( )
(
u(t)) :
(4.124)
Alternatively, one an get (4.72), (4.74), and (4.124) by a dire t appli ation of Corollary 1.60 to the original problem using the original geometry indu ed on
L p (I; Rm ).
Uad
from
As a result, in the linear/ onvex ase, the nely relaxed problems admit a twofold understanding: either as auxiliary problems imposing a suitable geometry just for a
70
See, e.g., Buttazzo [161℄ where sequential weak lower semi ontinuity is demonstrated. However,
the oer ivity of the ost fun tion together with metrizability of the weak topology of
L p (I;
Rm
) on
bounded subsets implies the weak lower semi ontinuity, as well.
71
If the Hamiltonian is stri tly on ave, then even every solution to the nely relaxed problems has
this form; f. Proposition 4.27(ii).
72
This is obvious if
*0
# 0 be ause the Hamiltonian is then ane. For ¡ 0 we refer again, e.g., to *
0
Buttazzo [161℄, using also the des ent ondition (4.36) together with metrizability of the weak topology of
L p (I;
Rm
) on bounded subsets.
4.3 Optimal ontrol of nitedimensional dynami al systems
Ë 305
derivation of the pointwise maximum prin iple for the oarsely relaxed problems, or as usual ontinuous relaxation s hemes but with a spe ial property that some (or all) optimal relaxed ontrols are 1atomi ; f. also the proof of Corollary 4.39.
Remark 4.58 (Renement of FilippovRoxin's theory).
The onvexity ondition (4.79)
in the form (4.54a) an be, if used arefully in the proof in Theorem 4.29, ombined with the maximum prin iple (4.75) to weaken the onvexity ondition as
:r ò Rn : o ',f , for a.a.
( t ; r; R ( t ; r ))
Q(t ; r)
with
R(t ; r) # s ò M(t);
(
t ; r; s) ¢ 0 ;
t ò I with M(t) being an arbitrary estimate of the set of maximizers in (4.75), i.e. M(t) s ò S(t); Hy 0 ;
*
;
(
*
t ; s) # Hy 0 ;
*
;
(
*
t ; u(t)) :
This may sometimes enable us to get rened existen e results even if the onventional orientor eld
4.3.d
Q is non onvex; f. [341, 549, 561℄. Cf. also [699℄ for integral equations.
Approximation theory
ODE Further task we want to pursue is a numeri al approximation of (R POC ). Rather than
H
presenting a general theory, we want to demonstrate appli ations of the results developed previously in Se tion 3.5 to build one on rete (semi)dis retisation. The reader
an anti ipate that we hoose onvex inner approximation of the set of relaxed admissible ontrols
U ad . We will make only dis retisation in the tvariable but not svariable
so that we get in general only the s heme of Type II; see the lassi ation from Se tion 3.5. For s hemes of Type I see Remarks 4.62, 4.66 and 4.67 below. Let
d ¡ 0
be a time step. We will suppose
T/d
integer and use an equidistant
I . For d1 £ d2 ¡ 0, we also suppose d1 /d2 integer so d2 is a renement of the partition with d1 . Then we dene the p p m m proje tor Pd : Car ( I ; R ) Ù Car ( I ; R ) by partition of the time interval that the partition with
[
Pd h℄(t ; s) #
1 d
X I kd
h( ; s) d
if
t ò I kd :# [(k"1)d ; kd); k # 1; :::; T/d;
(4.125)
f. also Se t. 3.5.b. On this rather abstra t level, we will assume that there are some linear subspa e
V C p (Rm ) and a linear subspa e G su h that
G V H l(G V); G0 G L where l refers to the natural topology73 of
G0 #
73
℄ Gd d ¡0
with
G d # g ò L
I ; H is Ginvariant;
( )
(4.126)
Carp (I; Rm ) and
I ; :k # 1; :::; T/d ; gI d ò C(I kd )Ǳ ;
( )
k
Quite equally it would su e to onsider any ner topology, e.g. the topology indu ed by the norm
(4.117) or any ner form, if exists.
306
Ë
4 Relaxation in Optimization Theory
f. (3.164). Also, we will assume that
H
as well as its norm is ompatible with
Pd
from
(4.125) in the sense that74
Pd : H Ù H
:h ò H :
is a ontinuous proje tor
;
(4.127a)
lim h " Pd h H # 0 :
(4.127b)
d Ù0
H d # Pd H H . By Propositions 3.83(i) and 3.86(i), Pd Y H (I; Rm ) is p m a onvex, weakly*  ompa t subset of Y ( I ; R ). Supposing that S ( t ) forming the H
ontrol onstraints in U ad from (4.63b) is onstant,75 i.e.
:a.a. t ò I : we an easily see76 that even
S(t) # S0 ;
(4.128)
Pd U ad U ad . Also, it holds *
d1 £ d2 ¡ 0, so that the onvex in reases for d Ù 0; f. Proposition 3.83(iii). Pd2
p
*
Then we denote
whenever
here obviously
approximations
Pd1 Pd2 #
Pd U ad Y H (I; Rn ) p
*
To ensure the onvergen e of the approximate problems, we onsider problems without state onstraints, f. Remark 1.52. For simpli ity, we suppose that the state equation as well as the ost fun tional do not require any additional approximation to be handled ee tively.77 Thus we ome to the following (semi)dis retised relaxed problem:
d ODE (R POC ) H
T X ['
. 6 Minimize 6 6 6 6
0
> 6 subje t to 6 6 6 6 F
y Ǳ ℄(dt)%(y(T))
dy # f y Ǳ ; y(0) # y ; dt p ò Pd U ad YH (I; Rm ) ; y ò W 0
*
1;
q (I;
Rn
)
:
Proposition 4.59 (Convergen e of numeri al approximations). Let all the assumptions of Proposition 4.46 with # 0 together with (4.126)(4.128) be valid. Then (R
d ODE H P ) has a solution OC
lim min(RdH PODE ) # min(RH PODE ) ;
d Ù0
OC
(4.129a)
OC
Limsup Argmin(RdH PODE ) Argmin(RH PODE ) : d Ù0
74
In fa t, it su es to suppose
OC
OC
Pd H
(4.129b)
H be ause (4.125) ensures Pd h H ¢ h H so that Pd L H H ¢ 1 (
;
)
provided  H is the norm (4.117), whi h an be always supposed.
75
A generalization for pie ewise onstant
partition of
76
I is straightforward.
ÜÙ S(t) whi h is pie ewise onstant on some equidistant
Let us note that, supposing (4.128) and taking a sequen e {
dened by (3.165) also belongs to
77
t
Uad .
u k }kòN
Uad , the sequen e {u k } ò, ( ;
)
In fa t, this an be true only in very simple ases. In general, we need always a numeri al solver
for the system of ordinary dierential equation as well as a numeri alquadrature formula to evaluate the ost fun tional. We negle t this need to make the presentation learer.
4.3 Optimal ontrol of nitedimensional dynami al systems
Ë 307
d ODE
Proof. The existen e of a solution to (R H POC ) follows by the same arguments as for the
ODE
ase (R H POC ), see the proof of Proposition 4.46. Then the laimed onvergen e follows
d # . Also note that (1.101 ) is satised by Proposition 3.83(iv) be ause U ad is a B  oer ive onvex  omn pa ti ation of U ad sin e H ontains a oer ive integrand, e.g. ' y with y ò C ( I ; R ), dire tly by the arguments of Remark 1.51 even simplied as
f. (4.70h) and Proposition 3.67(ii).
V
If
from (4.126) is nitedimensional, the set of admissible relaxed ontrols
d ODE for (R H POC ) is, in fa t, a onvex subset of a nitedimensional linear variety78
Pd U ad *
and as su h, it an be implemented dire tly on omputers; then we fa e the approximation of Type I (in a
ord with the lassi ation in Se tion 3.5). In the opposite
ase, we have obtained the approximation of Type II and a further theoreti al eort
d ODE is needed to implement the semidis retised problem (R POC ) on omputers. Namely, H d ODE we have to pose and analyze the optimality onditions for (R POC ). Of ourse, thanks H to the onvexity of
Pd U ad , *
we are able to perform it in an entirely parallel way
ODE how it was done for (R POC ). Now, the maximum prin iple will involve the dis rete
Hamiltonian
h dy ò H d ;
*
H
dened by
h dy # Pd ;
*
*
 (
f y) " (' y) :
(4.130)
Proposition 4.60 (Maximum prin iple for approximate problems).79
Let all the assum
ptions of Proposition 4.50 together with (4.126)(4.128) be valid, and ( d ;
d ODE ). Then the pointwise maximum prin iple
y d ) be a so
lution to (R H P
OC
h dyd
;
*
Ǳ d (t) # sup h dyd sòS0
;
(
*
t ; s)
L1 (I), where the Hamiltonian h dy; ò H is given by (4.130) and n ) is a solution80 to the adjoint ba kward terminalvalue problem:
is valid in the sense of
ò Lq
*
(
I; R
(4.131)
*
d % f r y d Ǳ d (t) # ' r y d Ǳ d ; dt *
*
(T) # 0:
*
(4.132)
Proof. By the same arguments as in the proof of Proposition 4.50, realizing additionally that (R
d ODE H POC ) is un onstrained (so that one an re kon
0 # 1 and # 0), we get the *
*
adjoint terminalvalue problem (4.132) together with the inequality
: ò Pd U ad :
d
*
Pd (G V) # {g : I
" ; h yd £ 0 ;
*
(4.133)
Ù R pie ewise onstant on I} V is nitedimensional.
78
Let us note that the spa e
79
Su h kind of maximum prin iple has been also stated by Chryssoverghi and Ba opoulos in [212℄
and, for ellipti optimal ontrol problems, by Chryssoverghi and Kokkinis in [214℄ or, for a paraboli optimal ontrol problem, also in [213, Thm. 3.2℄.
80
Likewise (4.105), in general (4.132) is to be understood in the sense of distributions.
Ë
308 with
4 Relaxation in Optimization Theory
hy # ;
d
*
*
 (
f y) " (' y). Sin e Pd d # d and Pd # , we an obviously write *
*
" ; h y # Pd d " Pd ; h y # d " ; Pd h y # d " ; h dy ; ;
*
*
*
;
*
;
*
;
*
whi h allows us to rewrite (4.133) as
: ò Pd U ad :
d
*
;
(4.134)
*
d d * d , we have < ; h y; > # < ; P h y; > # < P ; h y; > for any ò ; ; d d n ). As a result, the inequality (4.134) holds even for every ò U , whi h gives ad d d the maximum prin iple < d ; h y ; > # max ò U ad < ; h y d ; >. This maximum prin iple d Sin e
p Y H (I;
Pd h dy # h dy
" ; h dyd £ 0 :
R
*
*
*
*
*
*
*
an be equally written in the form
T X 0
h dyd
;
*
Ǳ d (dt) # sup
u ò U ad
T d X h yd ; 0
(
*
t ; u(t)) dt :
Then one an just use Theorem 4.21(i) to lo alize this integral maximum prin iple to get eventually (4.131); note that, thanks to (4.70g) and (4.70h),
h dy ;
*
satises the de
Å
s ent ondition (4.36).
Corollary 4.61 (Chattering solutions to (RdH PODE OC )).
Let
the
assumptions
of
H be separable. Then: d ODE There exists a solution ( ; y ) to (R H P ) with being ( n %1)atomi .
Proposi
tion 4.60 be valid and (i)
OC
; y) solves (RdH PODE ), d m Ù fun tion h y ( t ; ) : R is katomi .
*
(ii) If (
OC
;
*
is the orresponding adjoint state and, for a.a.
ò Pd h dy; and
n%1) suitable onditions involving the integrands h # d ODE for 1 ¢ l ¢ n , solves (R POC ); we use here the hain of identities H satises (
y # Pd ; ' y # ; Pd (' y) # ; *
*
 (
U ad , whi h h l # [f y℄l
*
Proof. Likewise in the proof of Corollary 4.53, we an show that any
; '
t ò I , the
R a hieves its maximum at no more than k points, then
*
f y) " h dy ;
*
# ; f y Ǳ " 1; h dy Ǳ # ; f y Ǳ d " 1; h dy Ǳ d *
*
;
# d ;
*
 (
f y) "
*
h dy; *
;
*
# d ; Pd (' y) # d ; ' y :
Then the point (i) follows from Proposition 4.28 modied for our pie ewise homogeneous ase (details are omitted). Eventually, the point (ii) again follows from Proposi
Å
tion 4.27(i).
By means of Corollary 4.61(i) we an eventually implement the semidis retised
d ODE relaxed problem (R POC ) on omputers: we an onsider only su h pie ewise homoH geneous relaxed ontrols whi h are (
n%1)atomi ,
whi h form a nitedimensional
4.3 Optimal ontrol of nitedimensional dynami al systems
Ë 309
(non onvex) manifold.81 Then Corollary 4.61(i) ensures that this manifold ontains at
d ODE least one solution to (R H POC ).
Remark 4.62 (Dis retisations of Type I). Requiring the regularity of the data ' y pH ¢ 1 ( I ; C ( S )) L ( I ; C 2 ( S )) with ; ò (0 ; 1℄, we C and f y p H n ¢ C for H # W
S
an perform a full dis retisation by means of the proje tor P # P P ( f. Se ts. 3.5.b d d1 d2 1
;1
2
and 3.5. with
0
1
0;
1
2
repla ed by I ). For implementation see also Remark 4.66 below.
Remark 4.63 (Dis retisations of Type III).
One an think also about a dire t dis retisa
ODE tion of the original problem (POC ) by making the original set of admissible ontrols
Uad from (4.63b) smaller, e.g. Ud #
u ò U ad ;
uI d k
onstant
; k # 1; :::; T/d :
Then the relaxed problem serves only for an asymptoti al analysis. Alternatively, we ODE
an get su h approximation as the restri tion of the relaxed problem (R H POC ) to the
i
(generally) non onvex nitedimensional variety H (
U d ), and thus su h s heme an
be viewed as a dis retisation of Type III a
ording to the lassi ation from Se tion 3.5. Su h s heme is used quite often, alled a ontrol parametrization method.82 Error estimates
svariable of the test intef ; f. [661℄. Also the dimensionality of the resulting problem
require additionally a ertain smoothness in the
grands and thus also of
is, under the omparable rate of error of minima, higher than the (semi)dis retisation of Type II presented here. The smoothness requirements as well as the dimensionality are similar as if we would have made additional aggregation of the pie ewise homogenized Young measures ( f. Remark 4.62) to get a dis retisation of Type I.83
Remark 4.64 (Adaptive supportestimation strategy).
In omparison with Type II or
III, the dis retisations of Type I lead to problems whi h are onvex if the relaxed problem is onvex but have a very large number of variables, in parti ular if
onsidering again
Pd #
m Á 1. Yet,
PdI1 PdS2 from Se ts. 3.5.b and 3.5. , most of the oe ients in
the Youngmeasure representation (3.168) of the optimal relaxed ontrols are zero, be ause typi ally the optimal ontrols have a hattering hara ter and these Young measures have supports only at few points where the dis rete Hamiltonian (4.130) is maximized. In onvex relaxed problems, the maximum prin iple (4.131) is a su ient
ondition and, if one knows (at least approximately) the orre t Hamiltonian, i.e. the
orre t adjoint state
*
from (4.132), one an onsider only those points in
S where this
Hamiltonian is (approximately) maximized. This may de rease dramati ally the number of degrees of freedom. Of ourse, one does not know apriori the adjoint state
81
The dimension is
n
 (
n%1)  m
 (
. Yet,
T/d) be ause on T/d time subintervals we need to pres ribed n%1) ve tors from m and n mutually independent
the support of the underlying Young measure as (
R
weights appearing in the onvex ombination of Dira measures supported at these ve tors.
82
See, e.g., Goh and Teo [358, 758℄.
83
For a theoreti al omparison of su h s hemes, the reader is referred to [671℄.
310
Ë
4 Relaxation in Optimization Theory
d k # (d1 k ; d2 k ) with d1 k ¡ d1 k%1 , d2 k ¡ d2 k%1 , k # 1; :::, and, knowing the solution on d k dis retisation and the orresponding adjoint state , to a tivate only su h supporting points on the k d next rened d k %1 dis retisation where the fun tion h d ( x ; ) is maximized with some y one an exploit a series of su
essively rening dis retisations ;
;
;
;
;
;
*
; d
*
toleran e. This multilevel strategy dealing with the adaptively tuned dis retisation of Type I an thus be organised due to the ow diagram:
INITIALIZATION
BEGIN
CHOOSE INITIAL DISCRETIZATION
ACTIVATION
ACTIVATE THE GRID POINTS WITH GREAT VALUE OF THE HAMILTONIAN
OPTIMIZATION ROUTINE SOLVE THE DISCRETE PROBLEM CONSIDERING THE ACTIVE GRID POINTS
NO
IS THE MAX. PRINCIPLE SATISFIED AT ALL GRID POINTS ?
YES NO
CORRECTION TAKE GREATER TOLERANCE
FINAL DISCRETIZATION LEVEL
YES
END
REFINEMENT REFINE THE DISCRETIZATION AND TAKE THE ORIGINAL TOLERANCE
This was proposed in [179℄ and shown to have an ability to be more e ient than the plain dis retisation from Remark 4.62 for additively oupled problems like (4.112) whi h leads to a linearquadrati programming (LQP). In more general ases, some iterative solvers must be used for the dis rete problems. In this multilevel approximation strategy for the relaxed problems, a usage of an iterative linearprogrammingbased algorithm was devised in [89℄. Moreover, adaptive meshing of
as in [173℄ an advan
tageously be ombined with this adaptive supportestimate algorithm, f. [84℄.
4.3.e
Illustrative omputational simulations: os illations
ODE Let us illustrate the pre eding results on a on rete problem (POC ) with 3 1 y0 # ( 16 ; 10 ), T # 1, # 0, S(t) # f # ([f℄1 ; [f2 ℄) in the form
'(t ; r; s) # A3i#1(s " u i (t))2 % 2i#1 (r i " [yd (t)℄i )2 with
u1 (t) # 31 t % 23 ; yd (t) # (t "
1 4
)(
n # 2, m # 1,
R (i.e. no ontrol/state onstraints) and ' and
u2 (t) # 2t( 23 " t) % 13 ;
t"
3 4
)
; (t "
1 2
1 " t)(t "
)(
u3 (t) # "t 1 5
)
;
(4.135a)
4.3 Optimal ontrol of nitedimensional dynami al systems
f
t ; r; s) # r2 % s " 32 t4 % 95 t3 "
45
f
t ; r; s) # "r1 % (s " 1) % 43 t6 " 62 t5 " 45
t4 %
[ ℄1 ( [ ℄2 (
11
t2 %
32 15
t"
23 15
;
Ë 311 (4.135b)
2
23 15
373 270
t3 "
457 90
t2 %
t"
281 90
47 48
:
(4.135 )
p # 6 and q # 3 so that a relaxation H Car6 (0; 1; R2 ) is possible. The data (4.135) are
This problem satises the assumptions (4.70) with by hoosing a suitable subspa e
inf
#0
ODE (POC ) and we know the exa t solution to the relaxed problem ODE (R POC ). By Corollary 4.53 there is at least one 3atomi solution ( ) whi h is given H
hosen so that
; y
here, in terms of Young measures, by
# ( " t )Æ u1 t % ( " t " t )Æ u2 t % ( y(t) # (t " )(t " ); (t " )(1 " t)(t " )
t
where
2
1
5
6
3
( )
1
1
1
2
10
3
2
( )
1
3
1
1
4
4
2
5
is the Youngmeasure representation of
%
1 10
.
1 10
t % 21 t2 )Æ u3
t
( )
(4.136a) (4.136b)
This solution is even unique,84
whi h shows that, in parti ular, the estimate of number of atoms in Corollary 4.53
annot be improved. An illustrative omputational experiment al ulations for (R
d ODE H POC ) presented here
0; 1) with the timestep d # 2"
4
has used an equidistant partition of the time interval (
.
By Corollary 4.61, one an rely on the existen e of at least one 3atomi solution to
d ODE the semidis rete problem (R POC ) whi h an a tually be implemented on omputers. H
An initial (intentionally rather badly) guessed 3atomi ontrol sponding response85
y # () is shown86 in Figure 4.3.
as well as the orre
Three sele ted iterations are shown in Figure 4.3, namely the zero (initial) one, 20th, and the nal 430th one) obtained by a sequentialquadrati programming (SQP) optimization routine.87 For omparison, the (unique) optimal ontrol to the nonODE dis rete relaxed problem (R POC ) and the orresponding response is displayed by dotH
ted lines.
84
min(RdH PODE OC ) # 0, we have y # y d determined {u i (t); i # 1; 2; 3} lo alized uniquely. Then the onvex
As both terms in the ost fun tional must vanish if
uniquely and also the support
supp(
t)
ombination of the Dira s in (4.136a) is determined from (4.89) also uniquely be ause the ve tors
dyd /dt " f(t ; yd (t); u i (t)) with i # 1; 2; 3 form here a (3 , 2)matrix of a full rank. 85
Of ourse, this response has been omputed only numeri ally by an expli it Euler method but with
a small time step 1/3200 so that it an be onsidered numeri ally as exa t. For sti systems or for higher a
ura y, more sophisti ated methods (as e.g. RungeKutta) would have to be employed.
86 3
87
is displayed. a l i H (u l ) is displayed only by u l (t), while the weights a l (t) are not indi ated.
Only the support of the Young measure orresponding to
l#1
This means,
#
The SQP routine NLPQLD by S hittkowski [716℄ has been exploited. The timedis retisation of the
ontrol has used
d # 2"4 .
312
Ë
4 Relaxation in Optimization Theory
PSfrag repla ements
Starting point S(
1
; y) for the optimization routine
supp( ν )
0.6
y2
T =1
0 PSfrag repla ements
0
PSfrag repla ements
S1
A urrent point S(
1
T =1
y1
0.2
; y) after 20 iterations
supp( ν )
0.6
y
T=1
0
1
PSfrag repla ements
S1
The solution S(
1
T=1
0
PSfrag repla ements
y
2
0.2
; y) after 430 iterations
supp( ν )
0.6
T=1
0
y1
PSfrag repla ements
S1 Fig. 4.3:
T=1
0
y2
0.2 The starting point, and intermediate iteration, and the nal iteration (
; y) for the optimiza
RdH PODE OC ); only supports of the 3atomi Young measure but not the probability
tion routine solving (
distributions are displayed. Cal ulation and visualization: ourtesy of
Mar ela MátlováVítková
(for
merly Cze h A ademy of S ien es)
As the (unique) relaxed optimal ontrol for (R
ODE
H POC
) has, in fa t, an
representation and also the approximate relaxed ontrols ported.
d
L
Youngmeasure
remain boundedly sup
By a detailed analysis as in [661℄, one an see that the dis retisation error
min(RdH PODE OC )
" min(RH PODE OC ) is of the order at least O ( d ), whi h agrees with the experi
mental results, as shown on Figure 4.4.
4.3 Optimal ontrol of nitedimensional dynami al systems
Discretization error
G
u
Fig. 4.4:
a
−5
ra
te
e
d
The dis retisation error
ODE min(RdH PODE OC ) " min(RH POC )
n
10
Ë 313
sl
in dependen e on
o
p
e
O(
d.
d)
repla ements Figure 4.3 (nal iteration)
−6
10
−3
−4
2
−5
2
−6
2
Remark 4.65 (Warga's algorithm).
2
Time step
d
The implementation on basis of Corollary 4.61(i) is
d ODE H POC ). Warga [792℄ p n) proposed a steepest des ent algorithm whi h uses the onvex geometry of Y ( I ; H
not the only numeri al approa h to the semidis rete problem (R
R
but, after ea h iteration, the resulted relaxed ontrol (whi h may possibly not be implementable) is repla ed by a hattering ontrol exhibiting the same ee ts, i.e. driving the ontrolled system to the same state under the same ost, whi h an be already implemented on omputers; by the Carathéodory theorem 1.12 it an be shown that there is at least one (
n%2)atomi ontrol with this property.88 Thus in our ase, Warga's al
gorithm would handle 4atomi relaxed ontrols.
Remark 4.66 (Dis retisation of Type I). stru ture in terms of
r,
Sin e the data (4.135) have linear/quadrati
the dis retisation by the proje tor
Pd # PdI1 PdS2
outlined in
Remark 4.62 results to a linear/quadrati onvex mathemati alprogramming problem. Therefore, a global minimizer an be found by a nite solver,89 whi h is ertainly a great advantage resulted from the onvex stru ture kept in the fully dis retised problem. In more sophisti ated appli ations, taking su h advantage may be ome a ne essity; f. Example 7.3.13. Presented sample al ulations use again the time
d1 # 2"4 and additionally the proje tor PdS2 ( f. (3.167)) whi h makes dis retisation of S # ["1; 1℄ by 61 equidistant points so that the meshsize parameter is d 2 # 1/30; note that, for simpli ity, S has been restri ted now only on the interval ["1 ; 1℄ without hanging the set of solutions though the original S # R ould be also dis retised by, however, a nonequidistant mesh. The dis retisation proje tor
PdI1
with the time step
resulted solution is shown on Figure 4.5. One an see that the optimal solution of the
88
For appli ation of Warga's algorithm see Chryssoverghi and Ba opoulos [212℄ or, in ase of a
paraboli optimal ontrol problem, also Chryssoverghi [210℄.
89
The solution shown on Figure 4.5 has been al ulated by the a tivesetstrategy linearquadrati
programming routine QLD by S hittkowski [716℄.
314
Ë
4 Relaxation in Optimization Theory
dis rete problem need not be now threeatomi though, of ourse, in the limit it inevitably approa hes the (unique) optimal threeatomi solution.
1
supp( ν )
0.3
y
1
T=1
0
T =1
0
y2
PSfrag repla ements
S1
0.3
Fig. 4.5:
Cal ulated optimal solution (
and visualization: ourtesy of
d ; y d ) to (RdH PODE OC ), d
Mar ela MátlováVítková
Remark 4.67 (Coarser relaxations).
$ (d ; d 1
2)
# (1/16; 1/30). Cal ulation
(formerly Cze h A ademy of S ien es).
If one uses a su iently oarse relaxation, it may
happen that one gets immediately the nitedimensional onvex dis retisation (i.e. of Type I) when only applying the spatial dis retisation by
Pd
from (4.125). E.g., appli a
ODE tion of (4.121) to (POC ) with the data (4.135) leads to the hoi e
H # C(I)  {h0 } % L3
/2
I {h1 ; h2 }
( )
with
h0 (t ; s) # Ai#1 (s" u i (t)) ; h1 (t ; s) # s ; h2 (t ; s) # (s"1)2 : 3
2
(4.137)
Q : C(I) , L3 2 (I)2 Ù H : (g0 ; g1 ; g2 ) ÜÙ 2l#0 g l  h l , the adjoint mapping Q : H Ù r a(I) , L3 (I)2 makes the equivalen e Y Hp (I; Rm ) Ê w*b lr a I , L 3 I 2 m B i ( U ad ) where the embedding is dened by i ( u ) # Q i H ( u ); it is easy to see that i ( u ) # ( h 0 u ; h 1 u ; h 2 u ) be ause /
Considering
*
*
*
( )
i ( u ) ;
(
( )
g 0 ; g 1 ; g 2 ) # Q i H ( u ) ; ( g 0 ; g 1 ; g 2 ) # i H ( u ) ; Q ( g 0 ; g 1 ; g 2 ) *
#
2 T H X g l ( t ) h l ( t ; u ( t )) d t l #0 0
#
( h 0 u ; h 1 u ; h 2 u ) ;
(
g0 ; g1 ; g2 ) :
Also, if
ò Y H (I; Rm ) is pnon on entrating and has thus a Youngmeasure represen
tation
, then it is an easy exer ise to verify the formula
p
Q # (h0 Ǳ ; h1 Ǳ ; h2 Ǳ *
i.e., the parti ular omponents of
orresponding
hl.
Q *
)
;
are just the momenta of
(4.138) with respe t to the
We already met this ee t when aggregating Young measures by
h l # 1 v l ; f. the formula (3.168). p Pd Y H (I; Rm ) is nitedimensional be ause the linear spa e Pd H has a nite dimension, namely 3 T / d . However, the implementation of the resulted dis retisation is not so easy in general. For example, if S ( t ) $ S 0 is bounded, then
means of spe ial niteelement fun tions It is now lear that
*
4.3 Optimal ontrol of nitedimensional dynami al systems
from the formula (4.137) one an dedu e90 that
Q
*
L
with the subset in
ò L
I
3
( )
Pd U ad *
Ë 315
is anely homeomorphi via
of the form
I 3 ; :a.a. t ò I : (t) ò o[Pd h℄(t ; S0 ) ;
( )
Pd h℄(t ; s) # ([Pd h0 ℄(t ; s); [Pd h1 ℄(t ; s); [Pd h2 ℄(t ; s)). It should be however emphaS0 # [a ; b℄ ò R and u i pie ewise onstant on the partition T d of I , it is pra ti ally impossible to des ribe expli itly the onvex hull of 3 2 2 3 the urve [ P h ℄( t ; S 0 ) # {(A i #1 ( s " u i ( t )) ; s ; ( s " 1) ); a ¢ s ¢ b } in R . The di ulty of d this task depends essentially on the parti ular nonlinearities h l involved in a problem. A spe ial situation o
urs if m # 1 and the nonlinearities are polynoms then one
where [
sized that, even in the spe ial ase
an use the stru ture from Se t. 3.3.d.91
H still smaller than (4.137) would ause similar troubles: for example the H # L3 2 (I) {h1 ; h2 } would require the extension of the ost fun tional only by lower semi ontinuity, while still a smaller H would additionally require a multivalued Taking
/
hoi e
extension of the ontroltostate mapping.92
4.3.f
Illustrative omputational simulations: os illations and on entrations
We already saw that the on entration ee ts an be ombined with os illation ones. The situation from Figure 3.9 on p. 152 an be illustrated on the following Bolzatype optimal ontrol problem [459℄, enhan ing the Example 4.34:93
T
Minimize
J(y; u) :# X (2"2t% t2 )u(t) % y22 (t) dt % (y1 (T)"1)2 0
subje t to
dy # u ; y (0) # 0 ; dt dy max(0; u) min(0; u) # % ; y (0) # 0 ; dt 1" y $ (y ; y ) ò W (I; R ) ; u ò L (I) ; 1
1
2
2
1
with some
2
1;1
2
1
/ 7 7 7 7 7 7 7 7 ? 7 7 7 7 7 7 7 7 G
(4.139)
ò (0; 1). Obviously, y2 L2 I tends to be as small as possible. Let us note 2"2t%t2 ¡ 0 has the minimum at t # 1, whi h for es the minimizing ( )
that the polynom
sequen e to on entrate around this time instant. An example of a minimizing sequen e is the ontrol
90
One must use also the wellknown properties of integrals of multivalued mappings:
l PA S(t) dt for any A I measurable provided S : I Â±
Rm
P A
o S(t) dt #
is measurable, losedvalued, and inte
grably bounded; see Aubin and Frankowska [37, Thm. 8.6.4℄ for details.
91
The relaxation of optimal ontrol problem by algebrai moments has been used in [535, 606℄.
92
Cf. the approa h by Buttazzo [160, 161℄. For a omparison with the ner relaxation using the Young
measures see Mas olo and Miglia
io [519℄, or also [664℄.
93
For a dierent relaxation of the example (4.139) using res aling time was devised by Kamps hulte
[417℄, loosing onvexity of the relaxed problem, however.
Ë
316
4 Relaxation in Optimization Theory
t ò (1; 1 % "), t ò (1 % " ; 1 % "), otherwise ;
~/ " . 6 u " (t) # > "~/"
6 F
y2"
is small, namely
y2 L2
I
( )
(4.140)
if
0
for whi h the orresponding state while
if
y " # (y1" ; y2" ) has the omponent y1" as in (4.140) # O("). For ~ # 1/2, we get the inmum of the
problem (4.139) is 3/4.
R # " ; % of R and the analyti al solution to su h relaxed problem is ò r a 0; 1 , R , f.
The relaxed problem an take the twopoint ompa ti ation
S #
[
([
℄
℄ )
the notation (3.51), given by
Æ 1 Æ" dt Æ0 ~
(dtds) #
{
}
% (1")Æ % Æ ;
0
if
t#1
otherwise
(4.141a)
;
and the orresponding states given (a.e.) as
y1 (t) #
0 1"
if if
0¢t 1 ; 1 t¢1
y2 (t) # 0 :
(4.141b)
The numeri al approximation of Type I (i.e. onvex / nitedimensional) an be made by the proje tors from Remark 3.86 and Example 3.5. . The results from omputational implementation of su h dis retisation are presented in Figure 4.6.
Fig. 4.6: The (support
of the) DiPernaMajda measure and the orresponding response approximating
R#
d # 0:0125 of the time interval [0; 1:5℄ and the "; %℄ by 20 points. Cal ulations and visualization: ourtesy of Martin
the exa t solution (4.141) with the dis retisation dis retisation of
Kruºík
1
[
(Cze h A ademy S ien es).
Con entrations may o
ur not only at isolated points but they an be smeared out along the whole interval and, simultaneously, they an os illate like on Figure 3.10 on p. 157. This an be demonstrated on the following problem [459, 693℄:
u(t)2 1%u(t)4 0 %(y1 (t)" t)2 % y2 (t)2 dt 1
Minimize
subje t to
J(y; u) :# X 2 u(t) %
dy # u ; y (0) # 0 ; dt max(0; u) min(0; u) dy # % ; y (0) # 0 ; dt 1" y $ (y ; y ) ò W (I; R ) ; u ò L (I) ; 1
1
2
2
1
2
1;1
2
1
/ 7 7 7 7 7 7 7 7 7 7 7 ? 7 7 7 7 7 7 7 7 7 7 7 G
(4.142)
repla ements
Ë 317
4.3 Optimal ontrol of nitedimensional dynami al systems
0 1 and ¡ 0. Let us note that this fun tional is oer ive on L (I). Here again the minimum does not exist and we an see that the inmum is (1 " 2 /3), uk viz [693, Prop. 4.1℄. The example of a minimizing sequen e for # 0 is on Figure 4.7. 1
with
2
u
k
y 1; k
/k2
k
0
2
k
:::
3
k
:::
1
t
3/k
T
2/k
1")/k
(
y 2; k
1/k
2
0
"k Fig. 4.7:
One element of a minimizing sequen e {(
u k ; y k )}kòN
t
T for the problem (4.142) with
# 0 on
verging to (4.143).
Taking again R # [" ; %℄, the analyti al solution to the relaxed problem is òr a([0; 1℄ , R) is given by
(dtds) #
dt Æ" % (1")Æ % Æ dt Æ
0
if if
0
0¢ t ¢ 1" ; 1" t ¢ 1 ;
(4.143a)
and the orresponding states given as
t
y1 (t) #
0¢ t ¢ 1" ; 1" ¢ t ¢ 1
if
1"
if
y2 (t) # 0 :
(4.143b)
f. [693, Prop. 4.1℄. Although the data varies ontinuously in time, the optima relaxed
ontrol jumps at
t # 1"
if
1, whi h has
a similar hara ter as Tartar's broken
extremal, f. [575℄. The numeri al approximation of Type I an be made as in the previous example.
# 0:5
# 0:3
On Figure 4.8, we an see numeri al results for and , and the adaptive repla ements supportestimation strategy from Remark 4.64 has been used in [693℄.
% γS
supp(
d1
0 1−β
0.5
d )
1.0
t
d2
1.0
d2
d2
d2
# 1/32 y 2 t 1.0
y1
0.5
# 1/32
0
0.5
DiPernaMajda measure and the orresponding response approximating
d # 0:05 of the time interval [0; 1℄ and two dis reti# 1/10 (dash line) and 1/34 (full line) for R # ["; %℄. Cal ulations and visualization:
the exa t solution (4.143) with the dis retisation sations
# 1/10
# 1/10
" Fig. 4.8: The (support of the)
d2
ourtesy of
Martin Kruºík
1
(Cze h A ademy S ien es).
Ë
318 4.3.g
4 Relaxation in Optimization Theory
Optimal ontrol of dierentialalgebrai systems
A nontrivial appli ation of the relaxation method is to optimal ontrol of systems governed by dierentialalgebrai equations (DAE). We onne ourselves to so alled
ausal semiexpli it systems,94 also alled Hessenbergform DAEs. We will deal with the following optimal ontrol problem in the Bolza form:
DAE
(POC )
T X ' ( t ; y ( t ) ; w ( t ) ; u ( t )) d t
. Minimize 6 6 6 6 6 6 6 6 6 subje t to 6 6 6 6 6
0
% (y(T))
( ost fun tional)
dy # f(t ; y(t); w(t); u( t)) ; (state equation  dierential part) dt 0 # g(t ; y(t); w(t); u(t )) ; (state equation  algebrai part) y(0) # y ; (initial ondition) ( t ; y ( t ) ; w ( t ) ; u ( t )) ¢ 0 ; (state ontrol onstraints) u(t) ò S(t) (:a.a. t ò I ) ; ( ontrol onstraints) y ò W q1 (I; Rn1 ); w ò L q2 (I; Rn2 ); u ò L p (I; Rm );
> 6 6 6 6 6 6 6 6 6 6 6 6 6 6 F
0
1;
' : I , Rn1 , Rn2 , Rm Ù R, f : I , Rn1 , Rn2 , Rm Ù Rn1 , g : I , Rn1 , Rn2 , Rm Ù Rn2 , : Rn1 Ù R, y0 ò Rn1 , S : I Â± Rm a multivalued mapping, and : I , Rn1 , Rn2 , Rm Ù R are subje ted to ertain data quali ation introdu ed later, n 1 ; n 2 ; m ; £ 1, 1 ¢ p %, 1 q 1 ¢ % and 1 q 2 ¢ %. Of ourse, R is expe ted to be ordered by a one D so that the ondition ( t ; r; v; s ) ¢ 0 has a sense. The pe uliarity of that the w variable is not subje ted to a time derivative and may follow the speed or amplitude of the ontrol variable u . Thus this fast part of the state ( y; w ) may exhibit the fastos illation and the on entration ee ts like the ontrol
where
variable and the relaxation must be done arefully, ounting an impli it onstraint
w and u by respe ting the algebrai part g(t ; y(t); ; ) # 0 depending also on the slow part of the state, i.e. on y .95 This suggests to make a relaxation for ( w; u ) jointly. Yet, respe ting the mentioned impli it onstraint g ( t ; y ( t ) ;  ; ) # 0, this would lead to a onvex ompa ti ation depending on the state y be ause onvex ompa ti ations in w and u separately would need a biane extension of f ( t ; y ( t ) ;  ; ) and g ( t ; y ( t ) ;  ; ) and its joint ontinuity, whi h is rather overambitious, as we saw in relating values of
Se t. 3.6. , f. also Remark 4.72 below. Therefore, relying that no impli it restri tion is imposed on the ontrol
u (related
with the assumed ausality) and exploiting the underlying ODE, we translate the
94
y; w℄(t) does not depend on the derivatives t but on u(t) only. For some results in more general ases
The adje tive ausal here means that the solution [
du/dt ; : : : ; d k"1 u/dt k"1 (
)
(
)
at a urrent time
see [469℄.
# 0 ould be, in prin iple, treated as a state onstraint in
95 The algebrai part g ( t ; y ( t ) ; w ( t ) ; u ( t )) DAE ( OC ). Yet, su h approa h would forget the
P
impose any restri tion on the ontrol
spe ial hara ter of the algebrai part whi h does not
u and would yield a dierent optimality onditions than (4.150)
(4.152) or (4.161) below, involving an additional multiplier and would bring te hni al troubles with failure of ontinuity into
L type spa e usually required.
4.3 Optimal ontrol of nitedimensional dynami al systems
Ë 319
DAE results from the previous Se tions 4.3.a to (POC ). Assuming rather for notational sim
pli ity that all equations in the ontrolled system have the same index, we formulate the results for the index at most 3, exploiting the onditions (1.47), (1.48), or (1.52) here modied for the ontrol problem. In parti ular, (1.47) reads as
; w òCAR
;
p; q2
(
I , Rn1 , Rm ; Rn2 ) : g(t ; r; v; s) # 0 ã v # w(t ; r; s):
(4.144)
DAE ODE The problem (POC ) is equivalent to (POC ) in the sense that (
u ; y) ò Argmin(PODE OC )
when we use
ã
(
u ; y; w) ò Argmin (PDAE OC )
' # 'ODE , f # fODE , and
#
w # w(y; u)
with
ODE ODE in (POC ) and the exponent
q # q1 with
'ODE (t ; r; s) :# '(t ; r; w(t ; r; s); s) fODE (t ; r; s) :# f(t ; r; w(t ; r; s); s); ODE (
with
', f ,
t ; r; s) :#
(
(4.145)
(4.146a) and
t ; r; w (t ; r; s); s)
(4.146b) (4.146 )
DAE from (POC ); again, we will often omit the expli it dependen e on
and
t as well as write w(y; u) instead of Nw (y; u). Let us further dene the manifold M(t ; x) Rn2 , Rm where the admissible pairs (w; u) respe ting the algebrai state equation take values as
M(t ; r) :# (v; s) ò Rn2 , S(t); g(t ; r; v; s) # 0 :
(4.147)
One an apply the FilippovRoxin existen e theory as in Corollary 4.39, leading to:
Proposition 4.68 (Existen e of solutions to (PDAE OC ): index1 ase). Let (4.144) hold and ODE (POC ) with ' # ' ODE and f # f ODE from (4.146) and q # q and U ad from (4.63b) be 1
nonempty with
S measurable and losedvalued, and (4.70a,b,d,e,g,h)96 hold. Moreover,
let
:
a.a.
t ò I : r ò Rn1 :
QM (t ; r) :#
the orientor eld
' ( t ; r; v; s )%
R
QM (t ; r) R,Rn1 ,R
% ; f(t ; r; v; s) ; 0
(
t ; r; v; s)% D ò R1%n1 % ;
(
be losed onvex, where is
dened by
v; s) ò M(t ; r)
(4.148)
M from (4.147). Then the problem (PDAE OC ) has a solution.
Proof. 97 Let us note that the data quali ation allows us to use Proposition 1.37 so that, for a xed ontrol
96
f and g. E.g. (4.70g) an be granted by asf t ; r; v; s) ¢ (a1 % vq2 /q1 % sp/q1 )(1 % r) with some a1 ò L q1 (I).
Using (4.70) imposes ertain quali ation on the data
suming
97
u ò L p (I; R ), the initialvalue problem for the DAE in question has
w t ; r; s ¢ C 1 % (
)
(
r
) and (
Cf. [702, Proof of Prop. 1℄ for details.
Ë
320
4 Relaxation in Optimization Theory
a unique solution. We will employ the transformation (4.146) and aim to use CorolODE lary 4.39 for the transformed problem (POC ). Then (4.79) results, after the substitution
n1 # n and (4.146), and w # w(y; u), to
Q(t ; r) # '(t ; r; w(t ; r; s); s)% R%0 ; f(t ; r; w (t ; r; s); s) ; (
t ; r; w (t ; r; s); s)% D ò R1%n1 % ; s ò S(t)
# '(t ; r; v; s)% R% ; f(t ; r; v; s) ;
(
0
t ; r; v; s)% D ò R1%n1 % ;
v # w(t ; r; s); s ò S(t) # QM (t ; r) QM from (4.148). Then, the assumed onvexity of QM (t ; r) results to the onvexity Q, so that, by Corollary 4.39, the (now auxiliary) problem (PODE OC ) has a solution ( u ; y ). DAE Using the impli ation á in (4.145) and putting w # w( y; u ), we get a solution to (POC ).
with of
By ombining the transformation (4.146) with Corollary 4.36, one an formulate DAE also the maximum prin iple for (POC ).98 We will assume
g v (t ; r; v; s)
is a regular (
!! "1 ! !![ g v ( t ; r; v; s )℄ !!!
n2 ,n2 )matrix and
¢ (r);
wr (t ; r; s) " wr (t ; r ; s)!!!! ¢ ~ r " r
!! !!
1
with some
(4.149a) (4.149b)
ò C(Rn1 ), ~1 ò R, and with w from (4.144).
Proposition 4.69 (Maximum prin iple for (PDAE be ontinuous OC ): index1 ase).99 Let n n , ODE 1 1 with ), and let (4.144) hold and (POC ) with ' # ' ODE v s $ 0 and r ò C ( I ,R ; R and f # f ODE from (4.146) and q # q satisfy the assumptions (4.70) hold. Moreover, let DAE ( u ; y; w ) solve (POC ). Then there are £ 0 and òr a(I; R ) with ( ; ) #Ö 0, £ 0, (y)  # 0 on I su h that the following maximum prin iple holds in the sense of L (I):
0
( ; )
1
*
*
*
0
*
*
*
0
1
*
hy ;
0 ; *
(
*
t ; w(t); u(t)) #
where the manifold
max h v s òM( t ; y ( t )) y; 0 ; *
( ; )
(
*
t ; v; s)
for a.a.
tòI ;
(4.150)
M is from (4.147) and the Hamiltonian h y; : I , Rn2 , Rm Ù *
R is
dened by
h y (t ; v; s) :# (t)  f(t ; y(t); v; s) " 0 '(t ; y(t); v; s) *
;
with
ò BV(I; Rn1 ), *
*
*
solving, together with
ò L1 (I; Rn2 ), *
(4.151)
the adjoint terminalvalue
problem for the linear dierentialalgebrai system
98
There is a quite ommon belief in literature [349, 492, 555, 556, 722℄ that one an apply the standard
maximum prin iple to DAEs as usual. This is however not always true, as shown on ounterexamples [263, 702℄.
99
The maximum prin iple involving the manifold
M is from (4.147) has been used e.g. in [351, 702℄.
A dierent manifold and a dierent Hamiltonian has been devised in [350, Theorem 7.1.6℄.
Ë 321
4.3 Optimal ontrol of nitedimensional dynami al systems
d % f (y; w; u) % g r (y; w; u) # ' r (y; w; u) % r (y) dt r with (T) # r (y(T)) ; f w (y; w; u) % g w (y; w; u) # ' w (y; w; u) ; *
*
*
*
*
0
*
*
(4.152a)
0
*
*
*
(4.152b)
0
Rn1 valued measures on I . Moreover, in the un$ 0), one has # 1, # 0, and ò W I; Rn1 .
a tually (4.152a) holds in the sense of
*
onstrained ase (i.e. if
1;1
*
0
(
)
ODE
Proof. 100 We use Corollary 4.36 for the transformed problem (POC ) so that automati ally (
y; i
u
ODE H ( )) solves the relaxed problem (R H POC ), as exploited there. The Hamiltonian
(4.71) gives now
hode y (t ; s) # (t)  f(t ; y(t); w (t ; y(t); s); s) " '(t ; y(t); w (t ; y(t); s); s) *
;
(4.153)
*
while the adjoint equation (4.74) gives the adjoint terminalvalue problem
d # ['ODE ℄r " [fODE ℄r # ' r (y; w(y; u); u) % wr (y; u) ' w (y; w(y; u); u) dt " f r (y; w(y; u); u) " wr (y; u) f w (y; w(y; u); u) % r (y) *
*
*
for the adjoint state
*
*
(4.154)
ò W 1 1 (I; Rn1 ) with (T) # (y(T)). Using also the substitution *
;
*
w :# w(t ; y(t); s) turns the Hamiltonian (4.153) into the form h y (t ; v; s) # (t)  f(t ; y(t); v; s) " '(t ; y(t); v; s) *
;
*
for
(
v; s) ò M(t ; y(t)) :
(4.155)
Moreover, the maximum prin iple (4.106) then turns into (4.150), the initialvalue problem (4.14) together with
w :#
w(t ; y; u) and (1.47) gives just the DAE in (PDAE OC ),
and eventually (4.154) results to
d # ' r (y; w; u) % wr (y; u) ' v (y; w; u) dt " f r (y; w; u) " wr (y; u) f v (y; w; u) % *
with
(T) # (y(T)). *
*
*
r (y)
*
(4.156)
Due to (4.149a), there is
*
solving (4.152b), namely
R
#
*
[(
g w ) ℄"1 ((' v ) "
* "1 is in L (I; n2 ,n2 ) and, by (4.8a,e), (' ) " ( f v ) ). Let us note that, by (4.8g), [ g w ℄ w * 1 n * n 2 ), as laimed. By (1.47), we have 2 ) so that ertainly ò L 1 ( I ; (f ) ò L (I;
R
w
R
g(y; w(y; u); u) # 0 so that g r (y; w; u) % g v (y; w; u)w r (y; u) # 0. Using it for (4.152b)
multiplied by
wr , one gets
' v (y; w; u) wr (y; u) " f v (y; w; u) w r (y; u)
*
# g v (y; w; u) wr (y; u) # " g r (y; w; u) : *
Substituting it into (4.156) gives (4.152a). This shows that ( as laimed.
100
Cf. [702, Proof of Prop. 2℄ for details.
*
; *
*
) solves the DAE (4.152),
Ë
322
4 Relaxation in Optimization Theory
The ondition (4.144) often annot be fullled be ause the DAEs in question have a higher index. We will demonstrate the needed modi ations rst for the index2 ase,
g of C 1 (
assuming
)
 lass, and
g v (t ; r; v; s) # 0
and
g s (t ; r; v; s) # 0 ;
(4.157)
g # g(t ; r; v; s) depends r.101 Then, using the al ulations (1.49), like (1.50), we modify (4.144),
f. (1.48) for the former relation. In fa t, (4.157) means that only on
t
and
assuming
; w òCAR
;
p; q2
(
I , Rn1 , Rm ; Rn2 ) : v # w(t ; r; s)
ã G(t ; r; v; s) # 0 with G # g t % g r f :
(4.158)
ODE We again use the transformation (4.146) towards (POC ), and dene the manifold
M(t ; r) :# (v; s) ò Rn2 , S(t); G(t ; r; v; s) # 0 :
Proposition 4.70 (Optimal ontrol of index2 DAEs).
Let
(4.157)
(4.159)
and
(4.158)
hold.
Then: (i)
QM (t ; x), dened by (4.148) now with M from (4.159), is onvex x and a.a. t, then (PDAE OC ) has a solution. n If also the assumptions of Proposition 4.69 holds and let, for some : R 1 Ù R
If the orientor eld for all
(ii)
ontinuous,
Gv
be a regular ( n 2
,n
2 )matrix
! !
! with !!
"1
Gv
(
!
t ; r; v; s)!!!! ¢ (r)
(4.160)
G from (4.159). Then, for any (u ; y; w) solving (PDAE OC ), there are 0 £ 0 and ò r a(I; R ) with (0 ; ) #Ö 0, £ 0, (y)  # 0 on I and the maximum prin iple 1 1 n (4.150) is satised with M from (4.160), h y from (4.151), and ò W (I; R 1 ) 1 n 2 solving, together with ò L ( I ; R ), the terminalvalue problem for the following *
with
*
*
*
*
*
*
*
;
;
*
*
adjoint DAE:
d % f (y; w; u) % G r (y; w; u) # ' r (y; w; u) % r (y) dt r with ( T ) # r ( x ( T )) ; f v (y; w; u) % G v (y; w; u) # ' v (y; w; u) ; *
*
*
*
*
0
*
*
0
*
*
*
0
with
(4.161a) (4.161b)
G dened in (4.159).
Proof. It just opies the arguments for Propositions 4.68 and 4.69.
101
The ase
gv
#Ö 0 but singular would lead to various indi es in parti ular equations, whi h would
require a suitable ombination of the presented results.
4.3 Optimal ontrol of nitedimensional dynami al systems
The derivation of
w and M be omes ompli ated quite rapidly for in reasing
index. Let us show it only for the index Assuming
f
of
C1 (
)
Ë 323
 lass and
g
C2 (
of
)
3 whi h also appears in nontrivial appli ations.
 lass, we now have to suppose, in addition to
(1.48), also
gr fv $ 0
gr fs $ 0 :
and
(4.162)
The former ondition, already used as (1.52), means that the DAEs do not be ome a differential equation for the variable
v and again, the latter ondition implies the ausalG in (4.160) as
ity of the DAEs. Then, for the index3 DAEs as in (1.54), we modify
G # g tt % g rr f 2 % g r f r f % 2g tr f % g r f t :
(4.163)
Remark 4.71 (Singular perturbations). Repla ing the algebrai part of the ontrolled d system g ( y; w; u ) # 0 by the dierential equation " dt w # g(y; w; u) and adding an DAE ODE initial ondition for w , the problem (POC ) turns into the form (POC ). Assuming " ¡ 0
w being y. The asymptoti s for " Ù 0 is a natural (and very nontriv
small, this models dynami al systems with two time s ales, the evolution of qualitatively faster than of
ial) question. Su h problems are also alled singularly perturbed and for their limit analysis in the ontext of optimal ontrol see e.g. Z. Artstein [28, 3032℄.
Remark 4.72 (Dire t relaxation of (PDAE OC )). in the fast omponent
w
The mentioned on entrations/os illations
suggests to relax both
u
w
and
simultaneously. This would
lead to the relaxed problem
Minimize
T X 0
subje t to
' y Ǳ (dt) % (y(T))
dy # f y Ǳ ; y(0) # y ; dt 0 # g y Ǳ ; yǱ ¢ 0; q p y ò W q1 (I; Rn1 ); ò Y H2 (I; M(y)) ; 0
1;
;
/ 7 7 7 7 7 7 7
(4.164)
? 7 7 7 7 7 7 7 G
I; M(y)) denotes the onvex  ompa ti ation of {(w; u) ò L q2 (I; Rn2 ) , L p (I; Rm ); (w(t); u(t)) ò M(t ; y(t)) :a.a. t ò I} using H a separable linear subspa e of q p n m the anisotropi spa e Car 2 ( I ; R 2 ,R ) of Carathéodory integrand with dierent growth restri tion in variables w and s , ontaining the linear hull of ' y , f y , g y , and y. The pe uliarity now is that the set U ad (y) of admissible relaxed ontrols and fast
where
q ;p
Y H2
(
;
states (whi h annot be learly distinguished from ea h other) depends on the slow
y. For index1 DAEs, the algebrai part is dire tly ontained in the manifold M(y) q p g y Ǳ # 0 is fullled automati ally for any ò Y H2 (I; M(y)). Exploiting w the Nemytski mapping indu ed by w and its ontinuous extension N ( y; ) of the w Nemytski mapping N ( y; ) indu ed by w from Lemma 3.100, we may think about
state
and thus
;
Ë
324
4 Relaxation in Optimization Theory
an alternative relaxation
T X
Minimize
0
' y Ǳ Ǳ (dt) % (y(T))
dy # f y Ǳ Ǳ ; y(0) # y ; dt w # N (y; ); yǱ Ǳ ¢ 0; q2 p q n n2 m 1 1 yòW (I; R ); ò Y H2 (I; R ); ò Y H (I; R ) ;
subje t to
0
1;
/ 7 7 7 7 7 7 7
(4.165)
? 7 7 7 7 7 7 7 G
H2 Carq2 (I; Rm ) and H Carp (I; Rm ). Here we use semibiane extension f y Ǳ Ǳ and ' y Ǳ Ǳ from Remark 3.113.
with suitable
Example 4.73 (Me hani al des riptor systems).102
A on rete
example
of index3
DAEs o
urs in so alled me hani al multibody des riptor systems using redundant
oordinates being subje ted to some holonomi kinemati onstraints. For a general formulation see e.g. [735℄. Prominent appli ations are industrial robots and their optimal ontrol is typi ally related with traje tory planning. the following autonomous
ase:
M(q)
dq dq d q (0) # q ; % K q; # J(q) w % B(q; u) ; C(q) # 0 ; q(0) # q ; d t dt dt 2
0
2
1
(4.166)
q : I Ù Rn is a timevarying position (traje tory) of the robot, M : Rn Ù Rn,n n n n a regular mass matrix depending on q , the for e K : R , R Ù R involves Coriolis, n
entrifugal and possibly also fri tion ee ts, C : R Ù R des ribes kinemati onn k , n denoting the Ja obian matrix and straints assumed smooth with J :# C : R Ù R H :# C : Rn Ù Rk,n,n (used later) its Hessian, w : I Ù R is the orresponding
where
Lagrange multiplier expressing the rea tion for es to these onstraints, being in position of the fast variable. The ontrol
u:I ÙS
B :R ,R ÙR performed by the hoi e: n 1 # 2 n , n 2 # k , n
transmission fun tion
m
Rm a t as applied for es through a
n . Transformation to the DAE in
dq ; g ( t ; r; v; s ) # C ( r ) ; f ( t ; r; v; s ) $ [ f ; f ℄( r; v; s ) with dt f (r; v; s) # r and f (r; v; s) # M " (r )J(r ) v % B(r ; s) " K(r) : y # q;
1
1
2
1
1
2
2
1
1
1
DAE
(POC ) an be
(4.167a) (4.167b)
The onditions in (4.162) are fullled due to the following orthogonality:
g r f v # C r1 ; C r2
 [f1 ℄v ; [f2 ℄v
# (J; 0)  (0; M "1 J ) # 0 ;
g r f s # C r1 ; C r2
 [f1 ℄s ; [f2 ℄s
# (J; 0)  0; B s
# 0 :
Therefore, if
102
g dierentiated twi e with respe t to time, a
ording to (1.54) we get
For optimal ontrol of me hani al des riptor systems f. e.g. [350, 555, 556, 722℄.
Ë 325
4.4 Ellipti optimal ontrol problems
G(t ; r; v; s) # g rr f 2 % g r f r f % 2g tr f % g r f t % g tt (t ; r; v; s)
# H(r )r % JM " (r 1
of ourse, the term with
2 2
1
v % B(r ; s) " K(r) 1
J(r1 )
1)
Hr22 $ r H(r1 )r2 ò Rk 2
means [
# 0;
(4.168) 2
Hr22 ℄ # ni#1 nj#1 [ r2i r2 j C ℄r2 i r2 j ;
;
;
;
# 1; :::; k. Now the variable v appears in this expression and, if JM "1 J is regu
lar, we an express
w # w(y; u) # [JM "1 J ℄"1(y1 ) J(y1 )M "1 (y1 )K(y)" B(y1 ; u) " H(y1 )y22
: Hen e the manifold
M from (6.4), now timeindependent, an be expli itly obtained
in the form:103
M(r) :# (v; s) ò Rk,m : s ò S ; JM "1 (r1 )(K(r)" J(r1 ) v"B(r1 ; s)) # H(r1 )r22 : This proves that the DAEs (4.166) are indeed of the index 3. The ompatibility onditions (1.51) and (1.55) now read simply as position
q0
C(q0 ) # 0 and J(q0 )q1 # 0, i.e. the
initial
of the robot fullls the kinemati onstraints while its initial velo ity
q1
lies in the tangent spa e.
4.4
Ellipti optimal ontrol problems
In this se tion we want to demonstrate appli ations of the presented theory to relaxation of optimal ontrol problems where the state equation is governed by a nonlinear 2ndorder ellipti partial dierential equation in the divergen e form; su h nonlinearity is referred as quasilinear. We want only to illustrate basi te hniques, so that we
onne ourselves to a derivation of a orre t relaxed problem and orresponding optimality onditions; the stability analysis, approximation theory, as well as various
onsequen es for the original problem are more or less parallel to the Se tion 4.3 and are thus left as exer ises.
4.4.a
The original problem and its relaxation
We will onsider both distributed and boundary ontrol but, for simpli ity, we will not impose any state onstraints in most of the exposition; f. Remark 4.84. The boundary
ontrol will a t through a nonlinear onditions of the Robin type, though the reader
103
Here
we
assume
nondegenera y
supy1 òRn det(JM "1 J (y1 )) ¡ 0.
of
the
holonomi
onstraints
in
the
sense
that
Ë
326
4 Relaxation in Optimization Theory
an ertainly imagine a modi ation for the ase of Diri hlet boundary onditions, as well.104 For
being a domain in Rn , n £ 2, with a Lips hitz boundary
, we will deal with
the following optimal ontrol problem
ELL
(POC )
Minimize X ' ( y; u d ) d x % X ( y; u b ) d S (fun tional) . 6 6
6 6 6 6 subj. to div a(y; x y) # (y; x y ; ud ) on ; (state equation) 6 6 6
"n  a(y; x y) # b(y; ub ) on ; (boundary ondition) > 6 6 u (x) ò Sd (x) (: x ò
); (distributed ontrol onstraints) 6 d 6 6 6 6 ub (x) ò Sb (x) (: x ò ); (boundary ontrol onstraints) 6 6 1; q m ); u ò L p1 ( ; n1 ); u ò L p2 ( ; n2 ) ; y ò W (
; d b F
a.a. a.a.
R
R
R
xdependen e of a, b, , and n . Here, p ; p ò 1; %), q ò (1; %), and the Carathéodory mappings a : , Rm , Rm,n Ù Rm,n ,
: , Rm , Rm,n , Rn1 Ù Rm , b : , Rm , Rn2 Ù Rm , ' : , Rm , Rn1 Ù R, and : , Rm , Rn2 Ù R, and the setvalued onstraint mappings Sd : Â± Rn1 and S b : Â± Rn2 will be subje ted to ertain data quali ation spe ied later. The
ase n # 1 is ex luded be ause it would not t with the following relaxation s heme sin e the ( n "1)dimensional Lebesgue measure on is not nonatomi , whi h would
where, for notational simpli ity, we omit
1
2
[
ex lude usage of the theory from Chapter 3. ELL The problem (POC ) ts with the framework of Se tion 4.1 if one takes the data for
the problem (POC ) as follows:
Y # W 1 q ( ; Rm ) ; U # L p1 ( ; Rn1 ) , L p2 ( ; Rn2 ) ; ;
Uad # Ud , Ub
(4.169a)
with
Ud # ud ò L p1 ( ; Rn1 ); :a.a. x ò : ud (x) ò Sd (x) ;
X # W
1;
q
Ub # ub ò L p2 ( ; Rn2 ); :a.a. x ò : ub (x) ò Sb (x) ;
(
(u ; y) ò W
; R 1;
q
(u ; y);
J(u ; y) #
(
m
*
)
; R
(4.169b)
;
(4.169 )
m
*
)
dened for all
y ò Y by
y # X a(y ; x y): x y % (y; x y ; ud ) y dx % X b(y; ub ) y dS ;
X ' ( y; u d ) d x
% X (y; ub ) dS :
Let us re all that the state equation
(u ; y) # 0
(4.169d) (4.169e)
with
from (4.169d) is a so alled
ELL weak formulation of the state equation with the boundary ondition from (POC ).
104
Many authors dealt with ellipti optimal ontrol problems, though mostly in lesser generality
(typi ally with bounded ontrols and/or linear highestorder term). E.g., we refer to Alibert and Raymond [17, 18℄, Bonnans and Casas [131, 132℄, Bonnans and Tiba [135℄, Buttazzo [160℄, Buttazzo and Dal Maso [163℄, Casas [180, 181℄, Casas and Fernández [183185℄, Lions [494℄, Lou [497, 498℄, Ma kenroth [508℄, Raitums [625, 626℄, et .
4.4 Ellipti optimal ontrol problems
Ë 327
ELL Of ourse, the problem (POC ) need not have any solution in general, and thus a
relaxation is desired. To exploit the theory of ellipti equations from Se tion 1.4.b, we will impose the following quali ation on erning the data
a, b, and . As for a, we
again assume (1.63a) and (1.65a). We will use Lemma 3.101 about semiane ontinuous extension for the following Nemytski mappings
N ' : L q ( ; Rm ) , L p1 ( ; Rn1 ) Ù L1 ( ) ;
(4.170a)
N : L q ( ; Rm ) , L p2 ( ; Rn2 ) Ù L1 (
(4.170b)
*
)
;
N b : L q ( ; Rm ) , L p2 ( ; Rn2 ) Ù L(q ")
; Rm ) ; N : L q ( ; Rm ) , L q ( ; Rm,n ) , L p1 ( ; Rn1 ) Ù L q
(4.170 )
(
*
(
*
R
") ( ;
m
(4.170d)
)
¡ 0; let us remind the notation (1.42) and (1.40) now for q and q, respe
tively. Let us note that that the mapping N in (4.170d) works with the state argument ( y; x y ) from an anisotropi spa e with dierent exponents q and q . A
ording to *
with some
*
(3.192), we an formulate the basi growth/ oer ivity and ontinuity properties
sp1 ¢ '(x ; r; s) ¢
'(x ; r; s) " '(x ; r ; s) ¢
sp2 ¢ (x ; r; s) ¢
1(
(x ; r; s) " (x ; r ; s) ¢
b(x ; r; s) ¢
b(x ; r; s) " b(x ; r ; s) ¢
(
q " )
(
x ; r; ; s) ¢
q
*
x) % Cr
q
q
(
*
*
(4.171b)
q
/
%
/(
q " )
(4.171e)
*
(
(
% Cq
x) % Cr
q
*
/(
q
*
(4.171d)
/
q
q q
r " r ;
;
x) % Crq "1 % C r q "1 % Csp2
(
(4.171 )
x) % Crq "1 % C r q "1 % Cs1 q
q "1" ") (x) % Cr
*
(4.171a)
"1 % C r q "1 % Cs1/q r" r ;
*
(
*
x) % Crq % Csp2 ;
x ; r; ; s) " (x ; r ; ; s) ¢
(
(
q
x) % Crq "1" % Csp2
(
x) % Crq % Csp1 ; *
1(
")
% Csp1
r " r ;
(4.171f)
") ;
(4.171g)
/(
q
*
"1 % C r q "1 % Csp1 /q r" r *
*
x) % Cq"1 % C q"1 % Csp1
/
q
" ;
(4.171h)
¡ 0 arbitrarily small, C ò R, and with p ò L p ( ) or p ò L p ( ) with spe i exponents p . Moreover, in addition to the quali ation (1.63a) and (1.65a) of a , like
with some
(1.65b) and (1.66), we assume the uniform oer ivity in the sense
;" ¡ 0 :ud ò L p1 ( ; Rn1 ); ub ò L p2 ( ; Rn2 ); y ò W X a ( y; x y ) : x y
1;
q
(
; Rm ) :
% (y; x y; ud ) y dx % X b(y; ub ) y dS £ """"" y"""""W 1 q Rm " 1" ;
(
and the stri t monotoni ity (1.66), i.e. here
:ud ò L p1 ( ; Rn1 ) :ub ò L p2 ( ; Rn2 ) :y; y ò W
1;
q
(
; R m ) ; y #Ö y :
;
)
(4.172a)
Ë
328
4 Relaxation in Optimization Theory
X a ( y; x y )
" a( y ; x y ) : x (y" y ) % (y; x y; ud ) " ( y ; x y ; ud )(y" y ) dx
% X b(y; ub ) " b( y ; ub )(y" y ) dS ¡ 0;
(4.172b)
so that the ontrolled system has a uniquely determined response for ea h ontrol pair. Let us rst formulate the FilippovRoxintype existen e theory for the original ELL problem (POC ):
Proposition 4.74 (Existen e for (PELL OC )).
Let (1.63a), (1.65a), (4.171), and (4.172) hold. Let
also the orientor elds
R% ; x; r; ; s ò R %m ; s ò Sd x : x ò ; Qb x ; r :# x ; r; s % R% ; b x ; r; s ò R %m ; s ò Sb x : xò ; m m , n . Then the optimal ontrol problem are onvex for any r ò R and ò R Qd (x ; r; ) :# (
)
' ( x ; r; s )% (
(
)
(
0
(
0
1
)
(
1
))
(
)
(
)
(
a.a.
a.a.
(4.173a)
)
(4.173b)
)
ELL
(POC )
pos
sesses a solution. The proof of the above assertion straightforwardly modies the arguments in the ELL proof of Theorem 4.29. For this, we need to onstru t a suitable relaxation of (POC ). To
this goal, we will pro eed by the routine way, parallel with the pre eding se tion. As we need to relax here both distributed and the boundary ontrols, we take here two separable linear subspa es
Hd Carp1 ( ; Rn1 ) supposing
Hd to be C( )invariant and Hb
U ad d # b lH ;
YHpd1 ( ; Rn1 )
i (U ) d ;Bd Hd d *
Hb Carp2 ( ; Rn2 ) ;
and
to be
and
C(
(4.174)
)invariant, and then put
U ad b # b lH ;
i (U ) b ;Bb Hb b *
YHpb2 ( ; Rn2 ); (4.175)
where
Bd
;R
L p2 (
and
Bb
L p1 ( ; Rn1 ) and U ad d , U ad b is onvex in Hd , Hb .
refer respe tively to the norm bornologies on
n 2 ). Thanks to the spe ial form (4.169b),
*
;
*
;
Again, we will onne ourselves to the ontinuous ane extensions with respe t to the ontrols, whi h will be guaranteed if the subspa es
Hd and Hb will be su iently
large. Like in Example 4.56, one an here take
Hd # spang( yx y) % g (' y ); y; y ò W 1 q ( ; Rm ); g; g ò C( ; Rm ); ;
Hb # spang(by) % g ( y ); y; y ò L
q
(
; Rm ); g; g ò C( ; Rm ) :
These spa es are separable if normed by the norms in
Car
p2
(
;R
Carp1 ( ; Rn1 )
(4.176a) (4.176b) and in
n 2 ), respe tively, f. Proposition 3.102. Of ourse, we an take any larger sep
Carp1 ( ; Rn1 ) and Carp2 ( ; Rn2 ) for Hd and Hb , respe tively. q ( ; R m ) Ù R by Then we dene the extended ost fun tional J : H d , H b , W
arable linear subspa es of
*
*
1;
J (d ; b ; y) #
X
[
' y Ǳ d ℄(dx) % X [ y Ǳ b ℄(dS)
(4.177)
4.4 Ellipti optimal ontrol problems
with on
' y Ǳ d and
mapping
and
y Ǳ b
Ë 329
being understood, if needed, in the sense of measures
, respe tively. Furthermore, one an dene the extended stateequation
: Hd , Hb , W 1 q ( ; Rm ) Ù W 1 q ( ; Rm ) *
*
;
;
*
by
y µ # X a(y; x y) : x y % ( yx y Ǳ d ) y dx % X (b y Ǳ b ) y dS :
´ (d ; b ; y);
We an see that
(4.178)
(d ; b ; y) # 0 is just equivalent to saying that y ò W 1 q ( ; Rm ) is ;
the weak solution of the boundaryvalue problem
div a(y; x y) # yx y Ǳ d "n  a(y ; x y) # b y Ǳ b
on on
; :
§
(4.179)
This leads us to the following relaxed optimal ontrol problem Minimize subj. to
X
[
' y Ǳ d ℄(dx) %
X
[
y Ǳ b ℄(dS) ;
/ 7 7 7 7 7
div a(y; x y) # yx y Ǳ d on ; "n  a(y; x y) # b y Ǳ b on ; d ò U d ; b ò U b ; y ò W q ( ; Rm ) 1;
with
U d YHd1 ( ; Rn1 ) p
and
U b YHb2 ( ; Rn2 ). p
(4.180)
? 7 7 7 7 7 G
Then our additional oer ivity and
monotoni ity assumptions (4.172) guarantees the orre tness of the resulted relaxation s heme.
Lemma 4.75 (Corre tness of the extended state problem). Let p ; p ò [1; %), q ò (1 ; %) (1.63a), (1.65a), (4.171eh), and (4.172) be satised, let H d and H b be from 1
(4.176). Then: (i)
The extended state problem (4.179) possesses for any
2
# (d ; b ) ò YHd1 ( ; Rn1 ) , p
YHb2 ( ; Rn2 ) a unique weak solution y # () ò W 1 q ( ; Rm ). p
;
W
1;
R
q ( ;
R
R
mapping : YHpd1 ( ; n1 ) , YHpb2 ( ; n2 ) Ù m ) thus dened is (weak*,weak) ontinuous if restri ted on the losure
(ii) The relaxed ontroltostate
of bounded sets.
d # i Hd (ud ) with ud ò L p1 ( ; Rn1 ) and b # i Hb (ub ) with ub ò L p2 ( ; Rn2 ), ELL then y # ( ) solves the original boundaryvalue problem in (POC ). In other words, p n p n 1 q 1 1 2 2 (i Hd , i Hb ) # where : L ( ; R ) , L ( ; R ) Ù W ( ; Rm ) denotes the
(iii) If
;
original ontroltostate mapping.
d ò YHd1 ( ; Rn1 ) and b ò YHb2 ( ; Rn2 ), there are sep n p n quen es { u d k } k òN and { u b k } k òN bounded in L 1 ( ; R 1 ) and L 2 ( ; R 2 ) su h that i Hd (ud k ) Ù d weakly* in Hd and i Hb (ub k ) Ù b weakly* in Hb , respe tively. To p
Proof. By the very denition of ;
p
;
*
*
;
;
prove the existen e of the solution to (4.179), we shall just pass to the limit with the solutions
yk
that orresponds to (
ud k ; ub ;
;
k ), whi h means ea h
y k ò W 1 q ( ; Rm ) is ;
the weak solution to the boundaryvalue problem
div a(y k ;x y k ) # (y k ; x y k ; ud k ) "n  a(y k ;x y k ) # b(y k ; ub k ) ;
on
;
;
on
:
¯
(4.181)
Ë
330
4 Relaxation in Optimization Theory
This means that, for any
X a ( y k ; x y k ) : x y
y ò W 1 q ( ; Rm ), it holds ;
% (y k ; x y k ; ud k )  y dx % ;
X
b(y k ; ub
;
k)
y dS # 0 :

(4.182)
ud k ; ub k ), our assumptions ensure the existen e of y k ò W 1 q ( ); see Proposition 1.41. The sequen e {y k }kòN is 1 q bounded in W ( ), whi h an be shown by putting y # y k into (4.182) and using the uniform oer ivity (4.172a). We used also that the sequen es { u d k L p1 ;Rn1 } k òN and { u b k L p2 ;Rn2 } k òN are bounded. Then, taking possibly a subsequen e (denoted, for By the lassi al theory, for any (
;
;
;
just one weak solution ;
(
;
(
;
)
)
simpli ity, by the same index), we an suppose that
yk Ù y
weakly in
W 1 q ( ; Rm ) : ;
(4.183)
¡ 0, we an rely on that the lowerorder part is ompa t in W 1 q ( ; Rm ) . Then, using the uniform monotoni ity (1.65a) of the highestorder part "div a(y; x y), we an improve the weak onvergen e (4.183) into the strong onverUsing (4.170 ,d,) with ;
*
gen e. Then it is easy to pass to the limit in the integral identity (4.182) to obtain
X a ( y; x y ) :
for any
x
y % ( y x y Ǳ d )  y dx % X (b y Ǳ b )  y dS # 0
(4.184)
y ò W 1 q ( ; Rm ). Thus we an see that y is the weak solution to the relaxed state ;
problem (4.169) and, by (4.172b), it is determined uniquely and (4.183) holds even for the whole sequen e. The statement (i) has been thus proved. One an easily verify that the whole pro edure works also for
p
dk ò YHd1
;
% ( ;
Rn1
)
; Rn2 ) with % ò R% arbitrary but xed. Hen e the point (ii) holds true, as well. Also the point (iii) is obvious. Å p2 k and b ò Y Hb ; % (
Proposition 4.76 (Corre tness of the relaxation s heme). Let p ; p ò [1; %), q ò (1 ; %), U ad be nonempty,105 let (4.174), be valid, and (4.171) Then: 1
(i)
2
The relaxed problem (4.180) has a solution.
d and b , i.e. every solution to (4.180), are p1  and p2 non on entrating, respe tively, and an be attained by a minimizing adELL 1 missible sequen es for the original problem (POC ) whi h have relatively L weakly
(ii) Every every optimal relaxed ontrols
ompa t energy. (iii) Conversely, a limit of every minimizing admissible weakly* onverging sequen e for ELL
(POC )
(embedded via i H d
, i Hb into Hd , Hb ) solves (4.180). *
*
Proof. First, let us noti e that the extended ost fun tional
J
in (4.180) takes the form
J (d ; b ; y) # # X b(y)  v dS
: v ò H ( ; Rm ) 2
d ò U ad ; ò Y H2 ( ; Rm , R ) :
/ 7 7 7 7
(4.188)
? 7 7 7 7 G
ELL The minimum of (4.188) exists and is ertainly below the inmum of (POC ). For the
ò Y H2 ( ; Rm , R ) satisfying < ; h v > # P b ( y )  v d S , there exists a sequen e {( y k ; u k )} k òN attaining weakly* in H and with y k solving the the boundaryvalue problem a ( y k ) # ( y k ; u k ) on . equality (i.e. no relaxation gap), it is important that, for any
*
This does not seem known, however; f. [710℄. Another appli ation might be in optimal ontrol of the NavierStokes equation with the onve tive term (
y  )y written as %div(yy). If % ¡ 0 is onstant, this allows 1 2 n y  v % %(y y):x y " (y; u) v dx # 0 on Wdiv ( ; R ); 0
for the very weak formulation P
;
f. p. 48 for the notation. Yet, again, the zero relaxation gap is not obvious.
;
4.4 Ellipti optimal ontrol problems
4.4.b
Ë 333
Optimality onditions in semilinear ase
The further task on erns the optimality onditions. The problem will be onsiderably simplied106 if we onne ourselves to to a semilinear system when onsidering
a lin
ear, i.e. [
a(x ; )℄ij #
n H a ijkl ( x ) kl k ; l #1
with
aòL
(
; R m,n (
2
)
)
:
(4.189a)
a(x) instead of a(x ; ). Assuming again the uniform monotoni ity a(x) will be assumed positive denite uniformly in x. We will thus naturally
onsider q # 2 and strengthen data quali ation (4.186) as:
Then we will write (1.65a),
p
1 2 m n1 m n2 ' òCAR Hd di ( , R , R ; R); òCAR Hb di ( ,R ,R ; R) ;
p
*
2 ;
;1
2 ;
;
")
p 1 ;(2
òCAR Hd di 2;
*
;
p
2 b òCARHb di 2 ;
;1
;
"
;(2
(
,R ,R
)
;
(
m
m,n
,R ; R n1
m
)
;
(4.189 )
, Rm , Rn2 ; Rm ) :
Without any loss of generality, the spa es
(4.189b)
(4.189d)
Hd
Hb
and
an be supposed separable
and suitably normed; e.g. we an always take respe tively the universal norms from
Carp1 ( ; Rn1 ) and Carp2 ( ; Rn2 ), f. Example 3.76. These data quali ation already enables us to formulate the optimality onditions in terms of the pointwise maximum prin iples both for the distributed and the boundary ontrols.
Proposition 4.80 (Maximum prin iples). Let p ; p ò [1; %), (4.174), (4.172) with q # 2, (4.189) hold, and let (d ; b ; y) be an optimal relaxed ontrol, i.e. a solution to (4.180). Besides, let 2 £ 3 and 2 ¢ 3 hold.107 Then the following integral maximum prin iples 1
2
*
hold:
y;
y;
hd Ǳ d ℄(dx) # sup X hd
ud òUd
X X
[
[
y;
*
hb Ǳ b ℄(dS) # sup *
ub òUb
where the distributed Hamiltonian
y;
hd
X
y;
hb
*
(
*
(
x ; ud (x)) dx ;
x ; ub (x)) dS ;
*
and the boundary Hamiltonian
(4.190a)
(4.190b)
y;
hb
*
are
given respe tively by the formulae
y;
hd
*
(
x ; s) # (x) (x ; y(x); x y ; s) " '(x ; y(x); s) ; *
106
The general quasilinear ase has been treated by Casas and Fernández [183℄.
107
This ondition omes from the requirement
(4.191a)
2 ¢ q used in Lemma 3.103, whi h sounds here ¢ q " and 2q ¢ q " . , We an see that this requirement an be fullled only for three n with n ¢ 3) or, if the boundary ontrol would not be (or less) dimensional problems (i.e. R
onsidered, for n ¢ 5. as
2q
*
*
Ë
334
4 Relaxation in Optimization Theory
y;
hb
*
(
x ; s) # (x) b(x ; y(x); s) " (x ; y(x); s) ; *
(4.191b)
ò W 1 2 ( ; Rm ) solving in the weak sense the adjoint boundaryvalue problem *
with
;
"diva x %[ y x y Ǳ d ℄ %[ r yx y Ǳ d ℄ # ' r y Ǳ d n  a x %[ y x y Ǳ d ℄ %[b r y Ǳ b ℄ # r y Ǳ b
*
*
*
*
on
;
/ 7 7 7 7 7
:
? 7 7 7 7 7 G
*
*
on
(4.192)
' and is satised and if Sd and Sb are measur and
Moreover, if the oer ivity (4.171a, ) of
able losedvalued, then also the following pointwise maximum prin iples on hold:
y;
y;
hd Ǳ d (x) # sup hd *
sòSd (x)
y;
y;