Lectures on mathematical control theory

The book is written on the basis of lectures read by the author at the Faculty of Mechanics and Mathematics of the Al-Fa

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Lectures  on mathematical control theory

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of nu n order be

t

positive-definite. Then control w ˜  L I  R k transfers the trajectory of the system (2.14) from a point x  R n to the point x  R n if and only if w t  /

^w ˜  L I  R

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(2.16)

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where O t  x  x B t ) t  t W  t  t >) t  t x  x @ N t  B t ) t  t W  t  t ) t  t 

function z t X  t  I  – solution of a differential equation A t z  B t X t  z t  X ˜  L I  R k 

(2.17) The solution of the differential equation (2.14), corresponding to the control w t  /  is determined by the formula y t z t X  O t  x  x  N  t z t X  X  X ˜  L I  R k  (2.18) where z

O t  x  x ) t  t W t  t W  t  t x  ) t  t W t  t W  t  t ) t  t x  N  t ) t  t W t  t W  t  t ) t  t .

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ZKHUH f y t  u t  t w t  t  I 1HFHVVLW\LVSURYHQ Sufficiency/HWWKHYDOXH J X  u  /HWVVKRZWKDWWKHSURFHVVGHVFULEHGE\ WKH GLIIHUHQWLDO HTXDWLRQ   XQGHU FRQGLWLRQV     LV FRQWUROODEOH ,Q IDFW WKH YDOXH J X  u   LI DQG RQO\ LI DOPRVW HYHU\ZKHUH IROORZLQJ HTXDOLW\ KROGVSODFH X t  O t  x  x  N t z t X f y t X  u t  t  t  I  ZKHUH y t X z t X  O t  x  x  N  t z t X  t  I /HW w t X t  O t  x  x  N t z t X f y t X  u t  t   ZKHUH y t X x   y t X x 1RZUHODWLRQV    FDQEHZULWWHQDV y t X A t y t X  B t w t  y t x  y t x  ,W IROORZV WKDW y t X x t t  x  u  x t x  x t x  7KLV PHDQV WKDW V\VWHP   LV FRQWUROODEOH XQGHU FRQGLWLRQV     6XIILFLHQF\ LV SURYHQ 7KH WKHRUHPLVSURYHG /HPPDLet the matrix W t  t !  , function f x u t , x  R n  u  R m , t  I , is continuously differentiable by variables x u  R n u R m , function  F q t X  T t x  T t x  N  t z t X  f y u t ,  where O t  x  x T t x  T t x  T t  B t ) t  t W  t  t  T t

y

B t ) t  t W  t  t ) t  t  q X  u z  z t  R k u R m u R n u R n ,

z  C t x  C t x  N  t z t , C t ) t  t W t  t W  t  t , C t ) t  t W t  t W  t  t ) t  t .

Then partial derivatives

wF q t >X  T t x  T t x  N t z t  f z  C t x  C t x  N  t z t  u t @ ,   wX wF q t  f u y u t >X  T t x  T t x  N t z t  f y u t @ ,  wu wF q t  f x y u t >X  T t x  T t x  N t z t  f y u t @ ,  wz wF q t  N  t  N  t f x y  u t >X  T t x  T t x  N t z t  f y u t @ .   wz t

>

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F q q t

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wF q t RI k  m  n u k  m  n RUGHU wq 

/HPPD Let matrix W t  t !  , function f x u t continuously differentiable by x u  R n u R m , t  I , and inequality is fulfilled F q q  t  F q q  t  q  q t  q  q  R k m n .  Then the functional (2.19) under conditions (2.20), (2.21) is convex. If function f x u t twice continuously differentiable by x u  R n u R m , t  I , and inequality is fulfilled * q t * q t t  , q q  R k m n , t  I ,  then the functional (2.19) under conditions (2.20), (2.21) is convex. 3URRI )RU DQ\ IL[HG t  I  UHODWLRQ   LV D QHFHVVDU\ DQG VXIILFLHQW FRQGLWLRQIRUWKHFRQYH[LW\RIDVPRRWKIXQFWLRQ F q t E\YDULDEOH q LH F Dq    D q  t d DF q  t    D F q  t 

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t

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d D ³ F q  t dt    D ³ F q  t dt D J X  u    D J X   u  

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wF q t satisfies the Lipschitz wq

condition on a variable q in the area of R N  N k  m  n , if wF q  'q t wF q t  d L 'q  wX wX

wF q  'q t wF q t  d L 'q  wu wu 



wF q  'q t wF q t  d L 'q  wz wz

wF q  'q t wF q t  d L 'q  wz t wz t

where Li const !  i  , 'q 'X  'u 'z 'z t . 7KHRUHP Let the matrix W t  t !  , function f x u t continuously differentiable by x u and partial derivative

wF q t satisfies the Lipschitz wq

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at any point X  u  L I  R k u L I  R m can be calculated by the formula

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ZKHUH l ±/LSVFKLW]FRQVWDQWIURP   PU >K @ ±SURMHFWLRQRISRLQW K RQDFRQYH[ FORVHGVHW U  7KHRUHP Let the conditions of Theorem 3 be fulfilled and, moreover, let, U – convex closed set in L I  R m , sequences ^X n `  L I  R k , ^un `  U   L I  R m are determined by the ratios (2.57), (2.58). Then:  numeric sequence ^J X n  un ` strictly decreases;  X n  X n o  , un  un o  when n o f . If, in addition, inequality (2.27) holds place, the set M X   u

^ X  u  L I  R

k



is limited below  sequences ^X n ` ^un ` are minimizing, i.e.  

`

uU J X  u d J X   u

OLP J X n  un

LQI J X  u 

J

no f

X  u  X

L I  R k uU ;

 sequences ^X n `^un ` weakly converge to the set U  where U X  u  X J X  u J LQI J X  u  U z ‡ , X u X

^

`

 The following estimate of the rate of convergence is true  d J X n  un  J d

m  n  m n

const ! 

 The controllability problem (2.51)-(2.53) has a solution if and only if J X  u

J



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IXQFWLRQ w t  /  7KHRUHPLet matrix W t  t !  . In order for the system (2.64)-(2.69) to be controllable, it is necessary and sufficient that the value J v  u x  x  where v  u  x  x  L I  R k uU u S  u S  solution of optimization problem (2.67)-(2.69). 7KHSURRIRIWKHWKHRUHPLVVLPLODUWRWKHSURRIRIWKHWKHRUHP W t  t !  vector function f x u t  x  R n  /HPPD Let matrix u  R m  t  I continuously differentiable by variables x u  R n u R m  function  

F q t _ v t  T t x  T t x  N t z t  v  f z t  v  C t x   C t x  N  t z t  v  u t  t _  q v u x  x  z t  v  z t  v  Then partial derivatives F v q t  Fu q t  F z q t  F z t q t are determined by

formulas (2.23) - (2.26), respectively, and partial derivatives  F x q t >T t  C t f y u t @>v  T t x  T t x  N  t z t  x  f y u t @  )RUPXODV     FDQ EH REWDLQHG GLUHFWO\ E\ GLIIHUHQWLDWLQJ WKH IXQFWLRQ F q t 'HQRWHE\ F q q t Fv  Fu  F x  F x  F z  F z t   q t  R k m n u I   /HPPD Let W t  t !  U t  L I  R m  S   R n  S  R n   convex closed sets. Function f x u t continuously differentiable by x u and inequality holds place  F q q  t  F q q  t  q  q ! t  q  q  R k m n   Then the functional (2.67) under the conditions (2.68), (2.69) is convex. 7KHSURRIRIWKHOHPPDLVVLPLODUWRWKHSURRIRIWKHOHPPD 'HILQLWLRQ Let's say that the derivative F q q t satisfies the Lipschitz condition by a variable q in the area R N  N k  m  n , if F x q t >T t  C t f x y u t @>v  T t x  T t x  N t z t  x  f y u t @







x







_ Fv q  'q t  Fv q t _ d L _ 'q _ _ Fu q  'q t  Fu q t _ d L _ 'q _ _ F z q  'q t  F z q t _ d L _ 'q _ _ F z t q  'q t  F z t q t _ d L _ 'q _ _ F x q  'q t  F x q t _ d L _ 'q _ _ F x q  'q t  F x q t _ d L _ 'q _

where Li const !  i  norm _ 'q _ _ 'v 'u 'x  'x  'z 'z t _  W t  t !  , function f x u t is continuously 7KHRUHP Let matrix x  u  R n u R m and partial derivative F q q t satisfies differentiable by variables the Lipschitz condition. Then the functional (2.67) under the conditions (2.68), (2.69) is Frechet differentiable, the gradient J c v u x  x J vc v u x  x  J uc v u x  x  J cx v u x  x  J xc v u x  x 

 L I  R k u L I  R m u R n u R n

H

at any point v u x  x  L I  R uU u S  u S X can be calculated by the formula k

J vc v u x  x J xc v u x  x

F v q t  t  B t \ t  J uc v u x  x t

³F

 x

q t  t dt  J xc v u x  x

t

F u q t  t 

t

³F

 x

q t  t dt 



t

where q t v t  u t  x  x  z t v  z t  v  z t v  t  I  is the solution of the differential equation (2.68), and the function \ t  t  I  adjoint system \

F z q t  t  A t \  \ t

t

 ³ F z t q t  t dt  t  I 



t

In addition, the gradient J c [  H satisfies Lipschitz condition __ J c [  J c [  __H d l  __ [  [  __ X  [  [   X 



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vn x n

vn  D n J vc vn  un  x n  xn  un PU >un  D n J uc vn  un  x n  xn @ PS > x n  D n J xc vn  un  x n  xn @ xn PS > xn  D n J cx vn  un  x n  xn @   

n    H  d D n d

  H  !  l   H 

ZKHUH l const !   FRQVWDQWRI/LSVFKLW]IURP   7KHRUHP Let the conditions of Theorem 8, the sequence ^vn `  L I  R k  ^un `  U  ^x n `  S   ^xn `  S are determined by the formula (2.76). Then: 1) numeric sequence ^J vn  un  x n  xn ` strictly decreases; 2) __ vn  vn __o  __ un  un __o  _ x n  x n _o  _ xn  xn _o  when n o f If, in addition, inequality (2.72) holds place, the set M v  u  x  x ^ v u x  x  X  J v u x  x d J v  u  x  x ` is bounded: ^vn ` ^un ` ^x n ` ^xn ` are minimizing, i.e. 3) sequences OLP J vn  un  x n  xn n of

J

LQI J v u x  x  v u x  x  X  X

4) sequences

V

^vn ` ^un ` ^x n ` ^xn ` weakly converging to the set ^ v  u  x  x  X  J v  u  x  x J LQI J v u x  x  v u x  x  X  V z ‡

5) the following estimate of the convergence rate is valid  d J v n  u n  x n  xn  J d

m  n  m n

const ! 

6) the controllability problem (2.64)-(2.66) has a solution if and only if J  7KHSURRIRIWKHWKHRUHPLVVLPLODUWRWKHSURRIRIWKHWKHRUHP &RQVLGHUWKHFRQWUROODELOLW\SUREOHP    ZKHQWKHILQDOPRPHQWRI WLPH t  LV QRW IL[HG t  IL[HG ,W LV QHFHVVDU\ WR ILQG WKH ORZHVW YDOXH t t  IRU ZKLFKWKHV\VWHP    LVFRQWUROODEOHLHWKHUHLVDFRQWURO u t U t  SRLQWV x  S  x  S  VXFK WKDW WKH WUDMHFWRU\ RI WKH V\VWHP   LQ WKH VKRUWHVW WLPHLVWUDQVIHUUHGIURPWKHVWDUWLQJSRLQW x  S DWWKHPRPHQWRIWLPH t WRWKH SRLQW x  S LQWKHVKRUWHVWWLPH t  t t ! t   7KXVWKHVROXWLRQRIWKHSUREOHPRIRSWLPDOVSHHGLVWKHIRXU t  u t  x  x  ZKHUH u t  x  x   VROXWLRQ  RI FRQWUROODELOLW\ SUREOHP     FRUUHVSRQGLQJWRWKHORZHVWYDOXH t RIHQGSRLQWLQWLPH /HWWKUHHEHIRXQG u t  x  x U u S u S  t >t  t @ t ! t IURPWKHVROXWLRQRI WKH FRQWUROODELOLW\ SUREOHP     ZKHUH t  t   NQRZQ TXDQWLWLHV &KRRVH t t   $FFRUGLQJWRWKHDERYHDOJRULWKPE\VROYLQJWKHRSWLPL]DWLRQSUREOHP     ZH ILQG WKH IRXU v

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 t  I

    v ˜  L I  R  u t U t  x  S   x  S       S Z t : t ^Z ˜  L I  R  J t d Z t d G t  ae t  I `      ZKHUH J t J  t J S t  G t G t G S t  VSHFLILHGFRQWLQXRXVIXQFWLRQV 7KHRUHP Let matrix W t  t positively defined. In order for system (2.77) - (2.80) to be controllable, it is necessary and sufficient that the value J [  , where [ v  u  x  x Z  X L I  R r uU u S  u S u :  optimal control of (2.81)-(2.84). 7KHSURRIRIWKHWKHRUHPLVVLPLODUWRWKHSURRIRIWKHWKHRUHP /HPPD Let matrix W t  t !  , functions f x u t  F x t  x  R m  t  I  continuously differentiable by variables x u  R n u R m , function z

>t  t @ 

k

* q t

 

 

F q t  _ Z  F y  u t _ _ v t  O t  x  x  N  t z t  v 

 f y  u t _  _ Z  F y  u t _  q

v u x  x  z t  v  z t  v  Z 

 R u R u R u R u R u R  O t  x  x T t x  T t x  t  I  k

m

n

n

n

S

 



Then partial derivatives * v q t

F v q t  *u q t

* x q t

F x q t  C t F y  t >Z  F y  t @

* x q t

F x q t  C  t Fx y  t >Z  F y  t @

* z q t

F z q t   F y  t >Z  F y  t @



* z t q t

Fu q t  *Z q t

>Z  F y  u t @

x





x

F z t q t   N  t Fx y  t >Z  F y  t @

where y t z t  C t x  C t x  N  t z t  v  t  I  7KH SDUWLDO GHULYDWLYHV   FDQ EH REWDLQHG E\ GLUHFWO\ GLIIHUHQWLDWLQJ WKH LQWHJUDQGIURP  HTXDOWR * q t   'HQRWHE\ * q q t *v  *u  * x  * x  * z  * z t  *Z  q t  R N u I   ZKHUH N k  m  s  n  /HPPD Let matrix W t  t !  , functions f x u  t  F x continuously differentiable by x u , U t  S   S  : t  convex closed sets and the inequality holds place  * q q  t  * q q  t  q  q ! t  q  q  R N   Then the functional (2.81) under conditions (2.82) - (2.84) is convex. The proof of the lemma is similar to the proof of Lemma 2 'HILQLWLRQ Let's say that the derivative * q q t satisfies the Lipschitz condition by a variable q in the area of R N  N k  m  s  n , if 





_ * v q  'q t  * v q t _ d L _ 'q _ _ * u q  'q t  * u q t _ d L _ 'q _ _ * x q  'q t  * x q t _ d L _ 'q _ _ * x q  'q t  * x q t _ d L _ 'q _ _ * z q  'q t  * z q t _ d L _ 'q _ _ * z t q  'q t  * z t q t _ d L _ 'q _ _ *Z q  'q t  *Z q t _ d L _ 'q _

where Li const !  i  _ 'q _ _ 'v 'u 'x  'x  'z 'z t  'Z _  7KHRUHP Let matrix W t  t !  , functions f x u  t  F x continuously differentiable by variables x u , and partial derivative * q q t satisfies the Lipschitz condition. Then the functional (2.81) under the conditions (2.82)-(2.84) is differentiable in the Frechet sense, the gradient J c [ J vc [  J uc [  J xc [  J xc [  J Zc [  L I  R k u L I  R m u R n u R n u L I  R S H

at any point [  X

L I  R k uU u S  u S u :  H can be calculated by the formula

J vc [ * v q t  B t \ t  J uc [ * u q t  J xc [

t

³*

x

q t  t dt 

t

J xc [

t

³*

 x

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t

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 n 

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b d · §b d ¨ ³ ³ lk [  W u [  W d[dW    ³ ³ lmk [  W um [  W d[dW ¸ ¸ ¨a c a c ¸ ¨ ¨ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜¸ b d ¸ ¨b d ¨¨ ³ ³ lnk [ W u [  W d[dW    ³ ³ lnmk [ W um [ W d[dW ¸¸ a c ¹ ©a c b d

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k

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§ t · ¨ f t M k t dt ¸ ³ ¨ t ¸ ¨ ¸ ¨ ˜˜˜˜˜ ˜˜˜˜˜ ¸ ¨ t ¸ ¨ f n t M k t dt ¸ ¨ t³ ¸ © ¹

t

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HTXDWLRQ   7KHRUHPLet the matrix T a b c d

b d

³ ³ K [ W K [ W d[dW



a c

of order N n u N n be positive defined. Then the general solution of the integral equation (4.76) has the form b d

u [ W K [ W T a b c d a  Z [ W  K [ W T a b c d ³³ K [ W Z [ W d[dW  

where Z [ W  L Q R m is an arbitrary function. 3URYHRIWKHWKHRUHPIROORZVIURPWKHRUHPV :HFDOFXODWHWKHIXQFWLRQ f t  Z E\IRUPXOD  f t  Z

b d

³³ / t [ W u [ W d[dW  t  I a c

a c

>t   t @ 

ZKHUH u [ W LVGHILQHGE\IRUPXOD  7KHQWKHGLIIHUHQFH u [  W  u [  W LVD VROXWLRQRIWKHLQWHJUDOHTXDWLRQ  

b d

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a c

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a c

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7KHRUHPLet the matrix The estimation is satisfied 'u [ W



T a  b  c  d

b d

t § t · ¨ 'f t M t dt  'f t M t dt ¸   k ³t n k ¸ ¨ t³  © ¹

be positive defined by formula (4.77).

³ ³ K [ W T a b c d 'a  



d[dW 

a c

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b d

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 N



a c

§ / t   [ W · ¨ ¸ ¨ / t  [ W ¸  b :HLQWURGXFHWKHPDWUL[HVDQGYHFWRUV K [ W ¨  ¸ ¨ ¸ ¨ / t  [ W ¸ © N ¹ ZKHUH K [  W LVPDWUL[RIRUGHU n N    u m  b  R n N  

§ f t · ¨ ¸ ¨ f t ¸ ¨  ¸  ¨ ¸ ¨ f tN ¸  ¹ ©



7KHQWKHDSSUR[LPDWHLQWHJUDOHTXDWLRQLVZULWWHQLQWKHIRUP b d

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b  







 

a c

7KHRUHPLet the matrix b d

T a b c d

³ ³ K [ W K [ W d[dW

a c

of order n N    u n N    be positive defined. Then the general solution of the integral equation (4.81) is defined by formula  

b d

u [ W K [ W T a b c d b  w [ W  K [ W T a b c d ³ ³ K [ W w [ W d[dW   



a c

where w [ W  L Q R m is an arbitrary function. 3URYHRIWKHWKHRUHPLVVLPLODUWRSURYHRIWKHRUHP /HW 'u [ W u [ W  u [ W  7KHIXQFWLRQ 'u [ W  L Q R m LVDVROXWLRQRIWKH LQWHJUDOHTXDWLRQ b d

³ ³ / t  [ W 'u [ W d[dW

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a c

ZKHUH ' f  t

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b d

³ ³ / t  [ W u [ W d[dW

f  t  Z 

a c

7KHRUHPLet the matrix b d

'u

L

T a  b  c  d !  

Then the estimation is held 

 ³ ³ K [ W T a b c d 'b d[dW 

a c

§ ¨t © 'f  t t

· ¸ ¹

t

where 'b 'b  'bn , 'bk ¨ ³ 'f  t M k t dt  ³ 'f  n tM k t dt ¸ , t

'f  t  'f  n t ,

tI .

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7KHSUREOHPVDUHVHW 3UREOHP (Controllability problem without restriction). Find a control v x  t  L  Q , which transfers the system (4.83), (4.84) from the initial state u x M x , x  I  , to the given final state u x T \ x , x  I  , at the time moment T , where \ x  L I is a prescribed function. 3UREOHP (Controllability problem with restriction). Find a control v x t  V , which transfers the system (4.83), (4.84) from the initial state u x M x , x  I  , to the given final state u x T \ x , x  I  , at the time moment T , where \ x  L I is a prescribed function. 3UREOHP (Controllability problem with minimal norm). Find a control v x  t  L Q with minimal norm, which transfers the system (4.83), (4.84) from the initial state u x M x to the state u x T \ x  3UREOHP(Optimal performance problem). Let v x t  V , u x T \ x . The time moment T is not fixed. Find a control v x t  V , which for the short time T transfers the system (4.83), (4.84) from the initial state u x M x , x  I , to the desired final state u x T \ x , x  I . 7KH LQWHJUDO HTXDWLRQV 6ROXWLRQ RI WKH HTXDWLRQ   ZLWK FRQGLWLRQV  WKURXJKWKHVRXUFHIXQFWLRQFDQEHUHSUHVHQWHGDV u x T



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ZKHUH § e  On a T W M [ · ¸ ¨  ¨ e On a T W M  [ ¸ LN [ W ¨ ¸ \ N ¸ ¨   ¨ e On a T W M [ ¸ N ¹ ©  

§ \  · ¸ ¨ ¨ \  ¸ ¨  ¸  ¸ ¨ ¨\ ¸ © N ¹

$SSO\LQJWRWKHLQWHJUDOHTXDWLRQ  WKHRUHPRI†ZHREWDLQ /HPPDThe integral equation (4.90) has a solution if and only if when T 

S

S  T 

³³L

N

[ W L N [ W d[ dW

 

of order N u N is positive defined. 3URYH RI WKH OHPPD IROORZV IURP WKHRUHP  RI † E\ VXEVWLWXWLQJ K t W  RQ L [ W   

/HPPDLet the matrix be S !   Then the general solution of the integral equation (4.90) is defined by formula v N [ W

T 

p [ W  L N [ W S\ N  L N [ W S ³ ³ LN [ W p [ W d[ dW 

 

 

where p [ W  L Q is an arbitrary function. Moreover, control v [ W with minimal norm in L  Q is equal to      v N [ W L N [ W S\ N   3URYHRIWKHOHPPDIROORZVIURPWKHRUHPSUHVHQWHGLQ† /HW v [ W  EH D VROXWLRQ RI WKH LQWHJUDO HTXDWLRQ   :H FDOFXODWH D T 

³ ³ G x [  T  W v

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[ W d[ dW \  x \  x

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'\ N  '\ N

 

ZKHUH '\ N



³ '\ M

n

§ '\  · ¨ ¸ ¨  ¸  ¨ '\ ¸ N ¹ ©

x dx  n  N 



/HPPDLet the matrix be 'v N [ W



T 

³³ L

N

S !  .

Then the estimation is held 

[ W S'[ N d[ dW  OLP 'v N [ W

 

N

N of

 

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­° ½  ° ®v [ W  L Q  ³³ v [ W d[dW d r ¾  °¯ °¿ Q

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T 

p [ W  L N [ W S ³ ³ LN [ W p [ W d[ dW  p [ W  L  Q   

:HLQWURGXFHDVHW M

­ ®w [ W  L Q  w [ W ¯

T  ½ p [ W  L N [ W S ³³ LN [ W p [ W d[ dW ¾    ¿

$VLWIROORZVIURPOHPPDVROXWLRQRIWKHLQWHJUDOHTXDWLRQ  KDVWKHIRUP v [ W L N S\  w [ W  w [ W  M  :HFRQVLGHUDQRSWLPL]DWLRQSUREOHPPLQLPL]HWKHIXQFWLRQDO T 

I N v w

³ ³ >v [ W  L

N

S\ N  w [ W @ d[ dW o LQI  

 

 



 

DWFRQGLWLRQV      v [ W  V  w [ W  M   /HPPD Let the pair v [ W  w [ W  V u M be a solution of the optimization problem (4.93), (4.94) at N o f  In order to the function v [ W  V be a solution of the integral equation (4.86), necessary and sufficiently, that I N v  w  at N o f . 7KXVIRUVROYLQJWKHFRQWUROODELOLW\SUREOHPLQWKHFDVH v [ W  V QHFHVVDU\ WRILQGDVROXWLRQRIWKHRSWLPL]DWLRQSUREOHP      *UDGLHQW RI WKH IXQFWLRQDO 2SWLPL]DWLRQ SUREOHP     FDQ EH VROYHG E\ FRQVWUXFWLQJ WKH PLQLPL]LQJ VHTXHQFHV ^vn [ W `  V  ^wn [ W `  M  ZKLFK FRQYHUJHV WR v N [ W V  w N [ W  M  DW n o f  ZKHUH OLP v N [ W v [ W  V  OLP w N [ W w [ W  M  N of N of 7KHRUHP Functional (4.93) under conditions (4.94) is continuously differentiable by Freshet, gradient of the functional I Nc v w IcN v w  I c N v w  L Q u L Q

 

at any point v w  V u M is equal to   >v [ W  L [ W S \ N  w [ W @  L Q , 7KHRUHP  Gradient of the functional I Nc v w  L Q u L Q satisfies to the Lipshitz condition, i.e. I Nc v  w  I Nc v  w d L v  v L  w  w L ,  IcN v w

I c N v w

>v [ W  L N [ W S\ N  w [ W @  L Q ,  

N





 v  w , v  w  V u M





, l const !  .



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Dw [ W  Ew [ W >Dp [ W  E p [ W @  L N [ W S ³ ³ LN [ W >Dp [ W    

 E p  [ W @ d [ d W  M



 7KLVLPSOLHVWKDW M LVDOLQHDUPDQLIROGLQ L Q  LI p [ W {   [ W  Q WKHQ w [ W  M &RQVHTXHQWO\OLQHDUPDQLIROG M LVVXEVSDFHLHFRQYH[FORVHGVHW 7KHRUHP Any element f [  W  L Q has an unique projection on the set M , moreover PM > f [ W @

T 

f [ W  L N [ W S ³ ³ LN [ W f [ W d[ dW  [ W  Q    

where

PM > f [ W @

is a projection of the point f [ W on M .  

 

3URMHFWLRQRIWKHSRLQW f [ W  L Q RQ V LVGHILQHG f [ W  ­  °°r ˜ f [ W  if f L ! r   L PV > f [ W @ ®    °  f [  W  if f r  d   L °¯







 

 &RQYH[ IXQFWLRQDO :H FRQVLGHU WKH IXQFWLRQDO   XQGHU FRQGLWLRQV  DVLWLVVKRZQDERYHWKHVHW M LVFRQYH[DQGFORVHGWKHVHW V LVFRQYH[ FORVHGVSKHUH&RQVHTXHQWO\WKHVHW V u M LVFRQYH[DQGFORVHG 7KHRUHP Functional I Nc v w on the set V u M is twice continuously differentiable by Freshet and convex. 0LQLPL]LQJVHTXHQFHV:HFRQVLGHUWKHRSWLPL]DWLRQSUREOHP     :HFRQVWUXFWWKHVHTXHQFH ^vn [ W `  V  ^wn [ W `  M E\WKHUXOH vn  [ W PV >vn [ W  D n IcN vn  wn @  n        wn  [ W PM > wn [ W  D n I c N vn  wn @  n        ZKHUH   H  d D n d

  H  !  ,QSDUWLFXODU H L  H

L  D n 

  H  L

  L

*UDGLHQW I Nc v w IcN v w  I c N v w  LV GHILQHG E\ IRUPXODV     L !  LVD/LSVKLW]FRQVWDQWRI  PRUHRYHU PV >˜@  PW >˜@ DUHGHILQHGE\UHODWLRQV     7KHRUHP Let the sequences ^vn [ W `  V , ^wn [ W `  M are defined by relations (4.101), (4.102).Then: 1) The lower bound of the functional I N v w is reached on the set V u M and I N v w LQI I N v w I N vn  wn at any fixed N ; v  w V u M 2) The sequence ^vn [ W  wn [ W `  V u M is minimizing, i.e. OLP I N vn  wn I N LQI I N v w  v w  V u M  n of 3) The sequence ^vn [ W  wn [ W `  V u M is weekly converges to the point v N

vN [ W  wN

weakly

weakly

wN [ W at n o f , i.e. vn o vN , wn o wN at any

fixed N ; 4) The estimation of the convergence rate is held  d I N vn  wn  I N d

5)

c  n   c const !   n



I N OLP I N vN  wN  , then the equation If I NOLP of N of





OLP vN [ W V

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3UREOHP Find a method of reducing the initial boundary value optimal control problem (6.95) - (6.98) to the initial optimal control problem with a free right end of the trajectory. 3UREOHP Construct minimizing sequences for the initial optimal control problem, prove its convergence, and obtain an estimate of convergence. 3UREOHPFormulate an algorithm for solving the simplest problem of the calculus of variations. 7KHHVVHQFHRIWKHSURSRVHGPHWKRGIRUVROYLQJWKHVLPSOHVWSUREOHPLVWKDW WKHVHWRIDOOFRQWUROVLVGHWHUPLQHGHDFKHOHPHQWRIZKLFKWUDQVODWHVWKHV\VWHP V WUDMHFWRU\ IURP WKH LQLWLDO VWDWH x WR WKH ILQDO VWDWH x DQG DOO NLQGV RI D SDLU u t  x t IRUWKHERXQGDU\YDOXHSUREOHP    6XFKDSSURDFKWRVROYLQJ DSUREOHPDOORZVRQHWRLPPHUVHVROXWLRQVRIWKHRULJLQDOH[WUHPDOSUREOHPWRWKH LQLWLDO RSWLPDO FRQWURO SUREOHP ZLWK D IUHH ULJKW HQG RI WKH WUDMHFWRU\ 7KH LQLWLDO SUREOHPLVVROYHGE\FRQVWUXFWLQJDPLQLPL]LQJVHTXHQFH 7KHQRYHOW\DQGSUDFWLFDOYDOXHRIWKHPHWKRGFRQVLVWVLQWKHIDFWWKDWILUVWRIDOO WKHVROYDELOLW\RIWKHERXQGDU\YDOXHSUREOHPLVSURYLGHGLQWKHIRUPRIH[WUDFWLQJDVHW RIDGPLVVLEOHSDLUV u t  x t DQGWKHVROXWLRQRIWKHVLPSOHVWSUREOHPLVSHUIRUPHGE\ VHDUFKLQJIRUDQH[WUHPDODPRQJWKHVHWRIDGPLVVLEOHSDLUV,WLVQRWHZRUWK\WKDWWKH DOJRULWKPIRUVROYLQJWKHSUREOHPLVIRFXVHGRQWKHXVHRIFRPSXWHUV 

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wF q t be satisfied to the Lipshitz i.e. wq

wF q  'q t wF q t _d L _ 'q _  wX wX wF q  'q t wF q t _  _d L _ 'q _  wz wz wF q  'q t wF q t _  _d L _ 'q _  wz t wz t const !  i  'q 'X  'z 'z t  _ 'q _ _ 'X  'z 'z t _ . _



Then the where Li functional (6.108) is differentiable in the Frechet sense, the gradient Jc X  L I  R at any point X ˜  L I  R is calculated by the formula Jc X

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t wF q t  t dt ³  wz t t





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 x u t  y Z t  R u R u I  D  >@

then the functional J X is convex.  

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)URP WKH DERYH UHVXOWV LW IROORZV   LI T u  p  v  v  x  x  d  X  ± LV D VROXWLRQRIWKHSUREOHPRIRSWLPDOFRQWURO    IRUZKLFK I T  WKHQ u u t  x  x  6  U u S  u S ± DGPLVVLEOH FRQWURO   IXQFWLRQ x t  t   x  t  I ± VROXWLRQ RI GLIIHUHQWLDO HTXDWLRQ   VDWLVILHV WKH FRQGLWLRQV x t  t  x x  x t  t  x  G t  t  I  IXQFWLRQDOV g j u ˜  x  x d c j  j  m  g j u ˜  x  x c j  j m   m  DQHFHVVDU\DQGVXIILFLHQWFRQGLWLRQIRUWKHH[LVWHQFHRIDVROXWLRQ WRWKHERXQGDU\YDOXHSUREOHP    LV I T  ZKHUH T  X  ±LVDVROXWLRQ RIWKHSUREOHP     IRUDYDOLGFRQWUROWKHYDOXHRIWKHIXQFWLRQDO   LVHTXDOWR t

³ F x t  u t  x  x  t dt

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SUREOHP    IRUWKHYDOXH t

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Lecture 30.&RQVWUXFWLQJDQRSWLPDOVROXWLRQ  &RQVLGHU WKH RSWLPDO FRQWURO SUREOHP     'HILQH D VFDODU IXQFWLRQ V t  t  I LQWKHIROORZLQJZD\ V t

t

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t

 



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§ Or · ¨ ¸ C t ¨ B t ¸ D t ¨ ¸ ¨ Om r ¸ ©  ¹ ZKHUH P P t V t  PP x 









7KHQWKHRSWLPDOFRQWUROSUREOHP    KDVWKHIRUP P P t J I u ˜  x  x o LQI       8QGHUFRQGLWLRQV P

A t P  B F PP  u x  x  t  C t f PP  u t  D f  PP  u x  x  t  

   

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w ˜  L I  R  w ˜  L I  R r  w ˜  L I  R    ] t P  T  ] t P  T      

 

   

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w t  w t  w t  < t W K t K  W   t

< t   t P  P   R t  t ³ < t   t B  t B  t < t  t dt  

B  C t  D  w t a

t

t

³< t W B

R t  t





W B  W < t W dW  R t  t

R t  t  R t  t  

t

/  t  P   P

B  t < t   t R  t   t a

§ B < t  t R t  t a · ¨ ¸ ¨ C < t  t R  t  t a ¸ ¨¨ ¸¸

 © D < t  t R t  t a ¹









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 B  < t t   t R  t   t < t   t

K  t

§  B < t  t R t  t < t  t · ¨ ¸ ¨  C < t  t R  t  t < t  t ¸ ¨  D < t  t R  t  t < t  t ¸      ¹ ©







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§ K t · ¨ ¸ ¨ K t ¸  ¨ K t ¸ ©  ¹

< t  t R t  t R  t  t P  < t  t R t  t R  t  t < t  t P   K  t

 < t  t R t   t R  t  t < t  t  t  I  

7KHRUHP /HW WKH PDWUL[ R t  t !  . Then control w t w t  w  t  w  t  L I  R

 r  m

transfers the trajectory of the system (7.69)-(7.71) from any

starting point P  R

 n  m

to any given final state P  R

 n  m

if and only if 

v t  / t  P   P  K t z t  v 

w t W  ^w ˜  L I  R  w t 

v ˜  L I  R  t  I ` 

w t W 

v  t  / t  P  P  K t z t  v 

^w ˜  L I  R r  w t

v  ˜  L I  R  t  I ` r

w t  W 

m

^w ˜  L I  R   w t



 



 

v  t  / t  P   P  K t z t  v 

m

v  ˜  L I  R  t  I `



 

ZKHUH v t v t  v  t  v  t  z t z t  v  t  I ±VROXWLRQRIDGLIIHUHQWLDOHTXDWLRQ z A t z  B v t  C t v  t  D v  t  z t       m

v ˜  L I  R  v  ˜  L I  R r  v  ˜  L I  R   





 

7KHVROXWLRQRIWKHV\VWHP    KDVDIRUP ] t

z t  v  /  t  P  P  K  t z t  v  t  I  

     7KHSURRIRIWKHWKHRUHPLVVLPLODUWRWKHSURRIRIWKHWKHRUHP  /HPPD /HW WKH PDWUL[  R t  t !  . Then the boundary value problem (7.65) ± (7.68) is equivalent to the following problem.   w t W   w t F P]  u x  x  t  t  I                 w t  W  w t f P ]  u  t  t  I         

f  P]  u x  x  t  t  I  

  p t  V t ^ p ˜  L I  R p t F P]  t  Z t d p t d M t  t  I `    z A t z  B v t  C t v  t  D v  t  z t  t  I      

w t  W   w t







s

m

v ˜  L I  R  v  ˜  L I  R r  v  ˜  L I  R    x  x  S  u S  u t U t  J  : d  *   

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t

t

t

³ F q t  t dt

J  v u  p x  x  d  J

³>_ w t  F P] t  u t  x  x  t _ 











 _ w  t  f P] t  u t  t _  _ w t  f  P] t  u t  x  x  t _ 





 

 _ p t  F P] t  t _ @dt o LQI 

XQGHUFRQGLWLRQV  ±  ZKHUH w t  W   w  t  W   w t  W   v v  v   v   q t v  v   v   u p x  x  d  J  z t  z t  1RWH WKDW WKH RSWLPL]DWLRQ SUREOHP      ZDVREWDLQHGRQWKHEDVLVRIWKHUHODWLRQV     wF q t satisfies the wq

7KHRUHP /HW WKH PDWUL[ R t  t !  , derivative

Lipschitz condition. Then:  IXQFWLRQDO  XQGHUFRQGLWLRQV    LVFRQWLQXRXVO\)UHFKHW GLIIHUHQWLDEOHWKHJUDGLHQWRIWKHIXQFWLRQDO J  T

J  v T  J  v  T  J  v T  J  u T  J  p T  J  x T  J  x T  J  d T  J  J T  

T X



v  v   v   u p x  x  d  J  X  m



L I  R u L I  R r u L I  R  u U u V u S  u S u * u : 

m

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H

r

m

u R  u R 

m

s

n

n

X  H  J  T  H

DWDQ\SRLQW T  X FDQEHFDOFXODWHGE\WKHIRUPXODV J  v T J  v T

wF q t  t  B \ t  J  v  T w v

wF q t  t wF q t  t wF q t  t  D \ t  J  u T  J  p T wu wp wv t t  wF q t  t wF q t  t J  x T ³  dt  J  x T ³  dt    wx wx t t 



J  d T

t

³

t

wF q t  t dt  J  J T wd 



wF q t  t  C \ t  wv 

t

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t

wF q t  t dt  wJ

ZKHUH\ t  t  I ±VROXWLRQRIFRQMRLQWV\VWHP t wF q t  t dt   ³  w z t t

wF q t  t  A t \  \ t wz

\



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__ J  T   J  T  __d l __ T   T  __ T  T   X  

7KHSURRIRIWKHWKHRUHPLVVLPLODUWRWKHSURRIRIWKHWKHRUHP n n n &RQVWUXFW WKH IROORZLQJ VHTXHQFHV ^T n ` ^v   v   v   u n  p n  x n  xn  d n   J n `  X   E\DOJRULWKP n n  PV  >v  D n J  v T n @ v 

n 

v

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v

n PV  >v   D n J  v T n @ un 

PV > pn  D n J  p T n @

pn  n  

x

n PV  >v   D n J  v  T n @

xn 

PS > x  D n J  x T n @ d n  

J n 

n 



PU >un  D n J  u T n @ PS > xn  D n J  x T n @ 

P* >d n  D n J T n @

P: >J n  D n J  J T n @

  H !  l l  H ^v  ˜  L I  R   __ v  __d E `

V

V m

^v  ˜  L I  R  __ v  __d E ` U

: ^J  RJ d J d E ` X 





 



n

 d Dn d

ZKHUH



d

const !  ^v  ˜  L I  R r  __ v  __d E `

V

m

^u ˜  L I  R m  __ u __d E ` * ^d  R  d t  _ d _d E `

V  u V  u V  u U u V u S  u S u * u :  H 

7KHRUHP /HW WKH FRQGLWLRQV RI 7KHRUHP  EH VDWLVILHG, X  - is bounded convex closed set, sequence ^T n `  X  is determined by the formula (7.86). Then:  QXPHULFVHTXHQFH ^J  T n ` VWULFWO\GHFUHDVHV __ T n  T n __o  ZKHQ n o f  ,IPRUHRYHU F q t LVFRQYH[IXQFWLRQE\YDULDEOH q WKHQ WKHORZHUERXQGRIWKHIXQFWLRQDO  LVUHDFKHGXQGHUWKHFRQGLWLRQV  ±   J  T inf J  T min J  T J    T X 

T X 

VHTXHQFH ^T n `  X  LVPLQLPL]LQJ lim J  T n J 

inf J  T  

nof

X  n 

T X 

 VHTXHQFH ^T n `  X  FRQYHUJHV ZHDNO\ WR D SRLQW T  X   n

n

weakly weakly o v  v   o v  ^T J  T J  inf J  T min J  T `  ZKHUH v  T X 

T X 





weakly o p  xn o x   xn o x  d n o d  J n o J ZKHQ v o v  un o u  pn  weakly

n o f  T

weakly







v  v   v   u  p  x   x  d  J  

 LI J  T   WKHQ RSWLPDO FRQWURO IRU WKH SUREOHP     DUH u  U 

x  S   x  S  DQGWKHRSWLPDOWUDMHFWRU\ x t P] t P> z t  v  /  t  P  P  K  t z t  v @ t  I  



 

ZKHUH

v





v   v   v   P 

d j t  j  m  c j

cj j

O  x   O

m

m   m `





 P

J  x   c  c  Q

m

^c  R   c j

cj  d j

VDWLVILHG WKH LQFOXVLRQ x t  G t DQG



OLPLWDWLRQV     J u  x  x J   WKHIROORZLQJHVWLPDWHRIWKHFRQYHUJHQFHUDWHLVYDOLG  d J  T n  J  d

c  n  c  n

const !  

7KHSURRIRIDVLPLODUWKHRUHPLVJLYHQDERYH $ PRUH YLVXDO PHWKRG IRU VROYLQJ SUREOHP     LV WKH PHWKRG RI QDUURZLQJWKHUDQJHRIDGPLVVLEOHFRQWUROV 7KHRUHP Let the conditions of the theorem 8 be satisfied, X  V  u V  u V  u U u V u S u S u * is bounded convex closed set, sequence ^T n `  X  is determined by the formula (7.86) except for the sequence ^J n `  : Then:  QXPHULFVHTXHQFH ^J  T n ` ^T n `  X  VWULFWO\GHFUHDVHV  __ T n  T n __o  ZKHQ n o f ^T n `  X    ,ILQDGGLWLRQWKHIXQFWLRQ F q t LVFRQYH[IXQFWLRQE\YDULDEOH q ZLWK IL[HG J  WKHQ VHTXHQFH ^T n `  X   ZLWKIL[HG J J LVPLQLPL]LQJ ɫɥ  T n o T  X  ZKHQ n o f J J    J  T inf J  T n min J  T n   X 

Tn

X 

Tn

IROORZLQJHVWLPDWHLVWUXH  d J  T n  J  T d

c  c n

const !  n  ^T n `  X   

7KHSURRIRIWKHWKHRUHPIROORZVIURP7KHRUHPIRUDIL[HG J  : J J   VHH   /HW T  X  EH D VROXWLRQ RI WKH SUREOHP       ZKHQ J J  : 7KHUHPD\EHFDVHV YDOXH J  T !   YDOXH J  T   1RWLFHWKDW J  T t  T  X    ,I J  T !  WKHQQHZYDOXH J ZHZLOOFKRRVH J J DQGLI J  T  WKHQ QHZYDOXH J

J 

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Lecture 31.%RXQGDU\SUREOHPVRIRSWLPDOFRQWURORIOLQHDUV\VWHPV ZLWKTXDGUDWLFIXQFWLRQDOZLWKRXWUHVWULFWLRQV   3UREOHP VWDWHPHQW &RQVLGHU WKH IROORZLQJ RSWLPDO FRQWURO SUREOHP PLQLPL]HWKHIXQFWLRQDO t

J x u

>

@

  x t Q t x t   x t M t u t  u t R t u t dt o LQI   t³

 

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A t x  B t u t  P t  t  I

     x t x  x t x  u x  L I  R       

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>t   t @ 

m

§ Q t M t · ¨¨ ¸¸ t   t  I >t  t @  © M t R t ¹ :H QHHG WR ILQG RSWLPDO FRQWURO u t  t  I  RSWLPDO WUDMHFWRU\ x t  t  I  $V N t

IROORZVIURPWKHUHVXOWVRI&KDSWHUIRUWKHH[LVWHQFHRIDVROXWLRQRIWKHSUREOHP    LWLVQHFHVVDU\DQGVXIILFLHQWWKDWWKHPDWUL[ t

T t   t

³ ) t  t B t B t ) t  t dt 





t

ZDV SRVLWLYHO\ GHILQHG ZKHUH ) t  W N t N  W  N t ± WKH IXQGDPHQWDO PDWUL[ RI x

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A t z  B t v t  z t

    7KH VROXWLRQ RI WKH GLIIHUHQWLDO HTXDWLRQ   FRUUHVSRQGLQJ WR WKH FRQWURO u t  U LVGHWHUPLQHGE\WKHIRUPXOD x t z t  O t  x  x  N  t z t  v  t  I       +HUH z

O t  x  x

  t  I  v x  L I  R m 

B t ) t   t T  t   t a  a

t

) t   t >x  ) t  t x @  ³ ) t   t P t dt  t

N  t

 B t ) t   t T  t   t ) t   t  T t   t T t   t  T t  t 

 

O t  x  x ) t  t T t  t T  t  t x  ) t  t  T t   t T  t  t ) t  t x   t

t

 ³ ) t W P W dW  ) t  t T t   t T  t  t ³ ) t  t P t dt   t

t

t

 ) t  t T t  t T  t   t ³ ) t  t P t dt  t

N  t ) t  t  T t   t T t   t ) t   t  t  I  /HPPD Let the matrix T t  t !   Then the boundary value optimal control problem (7.87)-(7.89) is equivalent to the following initial optimal control problem: minimize the functional 

t

J z x  z t v x

 ^> z t  O t  x  x  N  t z t  v @ Q t u  t³

u > z t  O t  x  x  N  t z t  v @  > z t  O t  x  x  N  t z t  v @ M t u u >v t  O t  x  x  N t z t  v @  >v t  O t  x  x  N t z t  v @ R t u

u >v t  O t  x  x  N t z t  v @` o LQI



Under conditions x

z

A t z  B t v t , z t

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