Dynamical Inverse Problems of Distributed Systems [Reprint 2014 ed.] 9783110944839, 9783110364132

Table of contents :
Introduction
Chapter 1. Problems of dynamical modeling in abstract systems
1.1. Reconstruction of inputs in dynamical systems. The method of auxiliary controlled models
1.2. Method of Stabilization of Lyapunov Functionals
1.3. Reconstruction of distributed controls in abstract systems. The case of instantaneous restrictions imposed on controls
Chapter 2. Approximation of inputs in parabolic equations
2.1. Reconstruction of distributed controls in nonlinear parabolic equations
2.2. Approximation of boundary controls in Neumann conditions and of coefficients of the elliptic operator
2.3. Reconstruction of intensities of point sources through results of sensory measurements
2.4. Approximation of boundary controls. The case of Dirichlet boundary conditions
Chapter 3. Approximation of inputs in parabolic variational inequalities
3.1. Dynamical discrepancy method
3.2. The role of a priori information in the problem of control reconstruction. The method of smoothing functional
3.3. Applications to inverse problems of mathematical physics and mechanics
Chapter 4. Finite-dimensional approximation of inputs. Examples
4.1. Reconstruction of boundary controls. The dynamical discrepancy method
4.2. Reconstruction of distributed control. The method of smoothing functional
4.3. Reconstruction of coefficients of the elliptic operators and boundary controls
4.4. The results of computer modeling
Bibliography

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INVERSE A N D ILL-POSED PROBLEMS SERIES

Dynamical Inverse Problems of Distributed Systems

Also available in the Inverse and Ill-Posed Problems Series: Theory of Linear Ill-Posed Problems and its Applications V.K. Ivanov.V.V.Vasin andV.P.Tanana Ill-Posed Internal Boundary Value Problems for the Biharmonic Equation MA. Atakhodzhaev Investigation Methods for Inverse Problems V.G. Romanov Operator Theory. Nonclassical Problems S.G. Pyatkov Inverse Problems for Partial Differential Equations Yu.Ya. Belov Method of Spectral Mappings in the Inverse Problem Theory V.Yurko Theory of Linear Optimization I.I. Eremin Integral Geometry and Inverse Problems for Kinetic Equations AKh.Amirov Computer Modelling in Tomography and Ill-Posed Problems M./H. Lavrent'ev, S.M.Zerkal and O.ETrofimov An Introduction to Identification Problems via Functional Analysis A. Lorenzi Coefficient Inverse Problems for Parabolic Type Equations and Their Application P.G. Danilaev Inverse Problems for Kinetic and Other Evolution Equations Yu.E. Anikonov Inverse Problems of Wave Processes AS. Blagoveshchenskii Uniqueness Problems for Degenerating Equations and Nonclassical Problems S.P. Shishatskii, A. Asanov and ER. Atamanov Uniqueness Questions in Reconstruction of Multidimensional Tomography-Type Projection Data V.P. Golubyatnikov Monte Carlo Method for Solving Inverse Problems of Radiation Transfer V.S.Antyufeev Introduction to the Theory of Inverse Problems A.L Bukhgeim Identification Problems ofWave Phenomena Theory and Numerics S.I. Kabanikhin and A. Lorenzi

Inverse Problems of Electromagnetic Geophysical Fields P.S. Martyshko Composite Type Equations and Inverse Problems A I. Kozhanov Inverse Problems ofVibrational Spectroscopy AG.Yagola, I.V. Kochikov, G.M. Kuramshina andYuA Pentin Elements of the Theory of Inverse Problems AM. Denisov Volterra Equations and Inverse Problems A L ßug/ige/m Small Parameter Method in Multidimensional Inverse Problems A.S. Barashkov Regularization, Uniqueness and Existence of Volterra Equations of the First Kind A.Asanov Methods for Solution of Nonlinear Operator Equations V.P.Tanana Inverse and Ill-Posed Sources Problems Yu.E. Anikonov, BA. Bubnov and G.N. Erokhin Methods for Solving Operator Equations V.P.Tanana Nonclassical and Inverse Problems for Pseudoparabolic Equations A.Asanov and ER.Atamanov Formulas in Inverse and Ill-Posed Problems Yu.EAnikonov Inverse Logarithmic Potential Problem V.G. Cherednichenko Multidimensional Inverse and Ill-Posed Problems for Differential Equations Yu.EAnikonov Ill-Posed Problems with A Priori Information V.V.Vasin andA.LAgeev Integral Geometry ofTensor Fields VA. Sharafutdinov Inverse Problems for Maxwell's Equations V.G. Romanov and S.I. Kabanikhin

INVERSE AND ILL-POSED PROBLEMS SERIES

Dynamical Inverse Problems of Distributed Systems

V.l. Maksimov

III УSP III UTRECHT · BOSTON

2002

VSP P.O. Box 346 3700 AH Zeist The Netherlands

Tel: +31 30 692 5790 Fax: +31 30 693 2081 [email protected] www.vsppub.com

© VSP 2002 First published in 2002 ISBN 90-6764-369-6 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

Printed in The Netherlands by Ridderprint bv, Ridderkerk.

Preface In this monograph, problems of dynamical reconstruction of unknown variable characteristics (distributed or boundary disturbances, coefficients of operators etc.) for various classes of systems with distributed parameters (parabolic and hyperbolic equations, evolutionary variational inequalities etc.) are discussed. The procedures for solving the problems axe established. They are based on the methods of the theory of feedback control in combination with the methods of the theory of ill-posed problems and nonlinear analysis. These procedures are oriented to the work in real time and may be realized in computers. The general constructions are illustrated on numerical examples. The book is destined for students, post-graduate students of physicalmathematical education, and specialists in optimization theory. Investigations, which results were included in the monograph, were partially supported by Russian Foundation of Basic Research (projects 01-0100566 and 98-01-00046), INTAS (project 96-0816), International Science and Technology Center (project 1293-99).

Contents Introduction Chapter 1. Problems of dynamical modeling in abstract systems 1.1. Reconstruction of inputs in dynamical systems. The method of auxiliary controlled models 1.2. Method of Stabilization of Lyapunov Functionals 1.3. Reconstruction of distributed controls in abstract systems. The case of instantaneous restrictions imposed on controls . . . . Chapter 2. Approximation of inputs in parabolic equations 2.1. Reconstruction of distributed controls in nonlinear parabolic equations 2.2. Approximation of boundary controls in Neumann conditions and of coefficients of the elliptic operator 2.3. Reconstruction of intensities of point sources through results of sensory measurements 2.4. Approximation of boundary controls. The case of Dirichlet boundary conditions

1

5 5 17 24 59 59 92 120 147

Chapter 3. Approximation of inputs in parabolic variational inequalities 157 3.1. Dynamical discrepancy method 157 3.2. The role of a priori information in the problem of control reconstruction. The method of smoothing functional 176 3.3. Applications to inverse problems of mathematical physics and mechanics 203

χ

V.l. Maksimov

Chapter 4. Finite-dimensional approximation of inputs. Examples 4.1. Reconstruction of boundary controls. The dynamical discrepancy method 4.2. Reconstruction of distributed control. The method of smoothing functional 4.3. Reconstruction of coefficients of the elliptic operators and boundary controls 4.4. The results of computer modeling Bibliography

213 213 223 229 248 257

Introduction Problems of reconstructing unknown parameters of objects and processes using available information are well-known in engineering and scientific research. Many problems of the kind axe posed as static ones and solved by static algorithms. In such problems, data allowed to be used in numerical solution algorithms are given a prion, the solution algorithms do not take into account changes in data, which might occur during the solution process, and the latter process is not viewed as the unique one; it can be repeated. Numerous examples of problems of the kind can be found among inverse problems in mathematical physics, problems of approximations of functions, problems of open-loop (program) control and observation, etc. However, it is often necessary to reconstruct unknown parameters dynamically, i.e., synchronically with an on-going physical process, or, as engineers say, "in real time". Data used in the reconstruction process may vary, "float" in time, and be, moreover, strongly past-dependent. A general problem of dynamical reconstruction of unknown parameters can be described as follows. A dynamical system operating on a time interval Τ = [ίο,0] is given. At each point in time, t, the system's state is represented as an element x(t) of a finite- or infinite-dimensional space X. A system's motion, i.e., the evolution of its state x(t), develops under some input disturbance u(·) starting from a given initial state XQ. It is required to reconstruct the input u(·) using observations of the motion x(-). Let us identify the system under consideration with an operator A transforming every admissible initial state XQ and every admissible realization of the input disturbance u(·) into a system's motion x(-) being a function of time; we assume that the operator A is single-valued. We stress that we deal with dynamical systems, i.e., the systems whose state histories do not depend on future evolutions of input disturbances. Formally, this property is reflected in the requirement that A is a Volterra operator. A wellknown class of systems of the kind is the class of systems described by linear

2

V.l. Maksimov

parabolic equations with boundary conditions of the Dirichlet type (for example). For such systems, time-varying inputs u(-) are traditionally viewed as controls. However, it(·) can represent noncontrollable external actions, i.e., act as noises, or dynamical disturbances, or inner variable parameters of the system, or have other physical interpretations. At any moment i, the observation result can, generally, be a function of the current state x(t), z(t) = C(x(t)), which is called the output of the dynamical system. In this monograph, we study the situation where the output values z(·) are given not precisely, i.e., the actual observation results £(') = £h{·) are connected with z(-) by the relation κ(£Λ(·), *(•)) < h. Here к is a nonnegative functional, and h is a level of an observation error. In this situation, the problem of constructing the operator B~l inverse to the "input-output" operator В : (XQ,U(·)) -4 Z(-) is not solvable precisely; it belongs to the class of ill-posed problems. The role of B~l is played by a certain operator Dh whose set of definition is wider than that of В - 1 ; namely, Dh is defined on the set of all functions £(·) representing admissible observation results (the set of measurements). The value Vh(·) of the operator Dfι at a function £(·) no longer equals the sought input u(-). However, we view Dh as a suitable approximation to the operator B~l if Vh(·) lies sufficiently close to u(·) for small h. The closedness of Vh(·) to it(·) is understood in the sense that the value p(u(·), Vh(·)) of a chosen nonnegative functional (the closedness criterion) is close to zero. For generality, we do not require that the original problem has the unique solution, i.e., we admit that the inverse operator B~l may not exist. In other words, we assume that every output z(·) can be generated by a set of inputs. We denote this set by U(z(·)). Let us suppose that we are not interested in finding all the inputs u(·) from U(z(·)) but interested in finding only those ones that are selected by a certain selection principle. In the theory of ill-posed problems, the minimization of a certain functional acts often as a selection principle. Following this approach, we assume that we have a functional ω(·) defined on the set of all inputs and reaching its minimum value on every set U(z(·)). Then the subset UM·))

= {«(•) G U(z(·)) : «(«(·)) = «,(.,}

is selected in every U(z(·)). We are interested in shifting Vh{·) closer to U*(z(•))·, more accurately, our desire is to make the value ß(vh, U,{z(·))) =inf{p(u(·), vh(·)) :

u(-)eUM-))}

Introduction

3

small for small h. Finally, we require that the operator Dh possesses the Volterra property that allows us to compute Vh(t) not later than at time t: if £i(r) = £2(1") for τ < t, then Vh,i{r) = ν/ι,2(ΐ") for τ < t, where VhA·) = тл·), vh,2(·) = Dhb(·). Thus, the basis problem of dynamical modeling of unknown parameters, which is analyzed in this book, consists in the following: construct a Volterra operator Dh, more precisely, a family of such operators depending on the accuracy parameter h, such that for any admissible output z(·) and any admissible results £(·) = ξ/Д·) of measurement of this output, the convergence 0Ы-), ад·)))0 as Л 0 is ensured. The above problem of constructing a family Dh falls into the scope of inverse problems of the dynamics of controlled systems. These problems consist in finding unknown inputs to the systems using observations of systems' outputs. Every input determines the unique motion of a system; usually, the inputs are either time-varying controls regulating the system, or system's initial states, or, in the general case, pairs composed of controls and initial states. The output may represent any available information on the controlled process, often such information is provided by signals on the system's trajectory (this situation is typical for practical problems). We note here that inverse problems of dynamics axe used for the design of controls realizing prescribed motions. The first publications on this subject, which dealt with systems described by ordinary differential equations, provided criteria for unique solvability of inverse problems under the assumption that controls are sufficiently smooth, and studied the issue of the continuity of controls with respect to the observed outputs (signals on trajectories). If outputs are observed with errors, the inverse problems of dynamics become ill-posed, and the question of constructing their approximate solutions becomes equivalent to finding appropriate regularizing operators (algorithms). Most researches on regularization address the "open-loop" setting of the problem: the regularizing algorithms process the entire history of the observation results (in this sense they axe a posteriory algorithms). The question of constructing dynamical (Volterra-type) regularizing algorithms for finitedimensional controlled systems was raised in Osipov, Yu.S. and Kryazhimskii, A.V., (1983). In this paper, a stable closed-loop (positional) algorithm reconstructing a minimum-norm control for a system affine in control with fully observable states was suggested.

4

V". I. Maksimov

For systems with distributed parameters, inverse problems treated within the framework of the "open-loop" setting were studied by many authors. Therefore, we do not overview this research axea in detail; we only point out several important lines of research. Many publications are devoted to the issues of existence, uniqueness and stability of solutions of inverse problems for systems with distributed parameters and to development of numerical solution methods. Rather often the linearization method as well as the scheme of Newton-Kantorovich are applied for solving inverse problems. It should be noted that the methods of optimization theory are also widely used in the theory of inverse problems. This approach goes back to Tikhonov, A.N. and Marchuk, G.I. To determine unknown parameters, the least square method is also actively applied. In the present monograph, we use the "closed-loop" (positional) approach for construction of dynamical regularizing algorithms for some classes of distributed systems. We illustrate general methods by examples and discuss problems of reconstructing distributed and boundary controls as well as coefficients in parabolic and hyperbolic (linear and nonlinear) equations and in parabolic variational inequalities. It should be noted that the method of positional control with a model lies, as a rule, in the basis of constructions we suggest. This is one of the most effective methods of the theory of positional control; its characteristic property is robustness to informational and computational errors. This method originally suggested by Krasovskii, N.N. and developed by Ekaterinburg's scientific school is used to construct stable procedures of dynamical modeling (reconstructing) of input disturbances uniquely determining a motion of a dynamical system with distributed parameters on the basis of measurements of current system's states. This monograph presents results of those studies only, in which the author's contribution is essential. Other algorithms and analytic methods for inverse problems of dynamics of distributed systems, which were developed within the framework of the above approach, can be found in the bibliography. The author thanks Yuri S. Osipov and Arkadii V. Kryazhimskii for their advise and recommendations, which essentially influenced the results of research presented in this monograph, and his colleagues in the Department of Differential Equations of the Institute of Mathematics and Mechanics (the Ural Branch of the Russian Academy of Sciences) for their help in preparing the monograph.

Chapter 1. Problems of dynamical modeling in abstract systems In this chapter we describe an approach of construction of regularizing algorithms for solving of inverse problems of dynamics for systems with distributed parameters. This approach is based on the combination of the Lyapunov function method and the principle of positional control with a model. It was suggested by Kryazhimskii A.V. and Osipov Yu.S. (1983) and further developed by many authors (see Bibliography). In this chapter, we suggest conditions sufficiently general for the case when it is convenient to take the copies of real systems as models for solving inverse problems. These are the problems of modeling unknown inputs given approximate measurements of outputs. The conditions we obtain in this chapter axe clarified for some classes of evolutionary systems. 1.1.

RECONSTRUCTION OF INPUTS IN DYNAMICAL SYSTEMS. THE METHOD OF AUXILIARY CONTROLLED MODELS

In this section, a general scheme for solving the problems of dynamical reconstruction of inputs through results of approximate measurements of outputs is described. The scheme is based on the method of auxiliary positionally controlled models well-known in the theory of guaranteed control. Further we fix ал interval Τ = where ίο < θ. The definition below is the modified Tikhonov definition (Tikhonov, A.N., 1939; see, also, Tikhonov, A.N. and Arsenin, V.Ya., 1978).

6

V. I. Maksimov Definition 1.1.1. An operator A:

XW χ.,.ΧΧΜ

->X(°\

where к Ε {1,2,... }, . . . , X^ k \ and tions from T, is called Volterra if for any *(1)(·),

y{1)(-)ex(1\

. . · , x{k4·),

are nonempty sets of func-

y{k4-)ex{k\

and

teT,

such that XCD(5) = y ( 1 ) ( s ) i

xW(s)

= yW(s) for all

sG[t0,t],

we have Α(χΜ(·),

. . . , *(•))(«) = A(yV(·),

... ,

for

all

se[t0,t].

Suppose X, U, XQ С X are nonempty sets of elements and XT0}E, UT0,E are nonempty sets of functions from Τ into X and U, respectively. Let A:

X0 Χ UTOF -> XT0TE

be an operator such that for any XQ G Xo the operator u(·) —>• A(Xo, U(·)) : Ut0,e Xt0,e is a Volterra operator; then for any u(·) € Ut0,e we have (Л(ж0,и(-)))(*о) = xo· The operator A is called a controlled dynamical system. Elements of the sets XQ and UT0LE are an initial state and a control (for the system A), respectively. The set of elements of the product Xo x Щ0 в is an input (of the system A).

The value A(XQ,U(·)) is a motion (of the system A) generated by an input (xo, «('))· The sets X and U are called a phase space and a space of controlled parameters (for the system A), respectively. Let Ζ be a nonempty set and С:

X-+Z

be an operator. We introduce an operator В : X0 χ Utofi -»· Ztо,*

Chapter 1. Problems of dynamical modeling in abstract systems where Zt(Sie is the set of all functions Τ В(х0,и(Ш

7

Ζ such that

= С(А(х0, и(Ш),

Finally, let us fix a functional P(v):

Ut0iexUt0>e^R+,

and, for a control υ(·) and an output z(·), we set ß(v(-),U(z(·)))

= inf{p(«(0.«(·)) : «(·) 6

ff.(*(·))}.

(1-1.2)

Further, the value β{ν(·), U„(z(·))) is called the p-error of control v(·) for output z(-) and the functional /£>(·,·) is called the criterion of error of approximation. The following basic definition follows from the theory of ill-posed problems (Ivanov et al., 1978; Tikhonov A.N. and Arsenin V.Ya., 1978). Definition 1.1.3. A family (D^) of Volterra operators acting from Ξίθι# into Ut0ie is called regularizing if for any output z(-) we have Umeup{j0(Z>Affc(·), U M ' ) ) ) ·

ξ Λ €Ξ(*(·),/ι)} = 0.

10

V. I. Maksimov

R e m a r k 1.1.1. Without loss of generality, we assume that the functional p(·, ·) is defined in the widen set U^g x U^ e (Ut 0i g С and the operator family (Dh) acts from Et0ie into U^g. However, mostly, it suffices to assume that Ut0te — ^t^fi· The problem in question is to construct a regularizing family of Volterra operators. If Ζ = X\ С = I (the identity operator), then the operators Д - 1 and A " 1 : Xtofi Utoj, Л - 1 ^ · ) ) = U(x(·)), A'1 = U.(x(·)) are Volterra operators for a sufficiently wide class of systems (this is proved below). Therefore, we should seek for a solution of the problem in the class of Volterra operators. The approach described below is based on the well-known principle of positional control: the principle of auxiliary controlled models. Its essence is as follows. We choose an auxiliary dynamical system (we call it a model). The initial state of the system WQ is given. The set of motions w(-) is given by a Volterra operator defined on pairs (£(·), u(·)), where £(·) is a result of measuring an output z(-) generated by a real input of the system A; and V(·) G Ut0ig is a control in the model. The initial state WQ of the model is given by the value ξ (to) of measuring at the initial moment to following a rule W fixed α priori (this rule is called a to-algorithm). The laws of forming a model control v(-) axe called strategies following the terminology of positional control theory (Krasovskii, N.N., and Subbotin, A.I., 1984). These strategies are identified with Volterra operators У, which, following the feedback principle, assign v(·) to a pair (£(•), w(·)) (measurement-motion of the model). A process (w(-),v(·)) (motion-control of the model), for the given pair (W, У), depends on a measurement £(·). Thus, the pair (W,3^) defines ал operator D acting on measurements £(·). Note that the operator Dh is a Volterra operator under some special conditions. We construct the regularizing family from such operators. The pairs (W, У) which define these operators are called modeling algorithms. Suppose that W is a nonempty set; Wo is a subset of W; Wto,e is a set of functions from Τ into W (W0 φ 0, Wto,e φ 0 ) , Ξ 0 = {({to)

•

£(·)β χ Uto

Wto,e

is an operator such that for any wo € Wo the operator (£(·)>v(")) M ( ι ν ο , ξ ( · ) , ν ( · ) ) is Volterra, and, for any £(·) G Eto>e and u(-) G Uto,e, we

Chapter 1. Problems of dynamical modeling in abstract systems

11

have (M(w0,

£(·), «(•)))(*)= «Χ>·

(1.1.3)

Definition 1.1.4. The operator Μ is called α model; the set W is called the phase space of the model, Wo is called the set of initial states of the model; the function u»(t) = Μ(ωο,£(·),«(·))(*) = w{t;to,wo,(to,t{-),vto,t{·))

eW,

teT (1.1.4)

is called a motion (phase trajectory) of the model. Definition 1.1.5. Any function W which may depend on h W = Wh:

(1.1.5)

wo = wh{to)=Wh(t{t0))eW0

is called an algorithm of choice of the initial state of the model. Definition 1.1.6. Let w0 e Wo, У : Et0ie x Wtoj Uto,e be a Volterra operator, t ET, and a measurement £t0,t(·) 6 ^t0,t· Then any pair (™toA-)'vto,t{·)) e Wt0,t x Uto,t such that wto,t(·) = Mt(wQ^to>t(-),vto>t{·)), is called a

(Y,(tQ,t('))~PROCESS

vto,t(·) =

with the initial state

yt(&0,t(-)>wto,t(·)) WQ.

Definition 1.1.7. A Volterra operator У : Ξ(θι# χ Wt0ß —>· Ut0,e is called an admissible strategy if for any WQ € WQ, t ET, and £t0,ö(·) e there exists a unique (^,6o,t(")) _ P r o c e s s with the initial state WQ. Definition 1.1.8. Any pair (W, where W is a io~ a lg° r ithm and У is an admissible strategy is called an algorithm of modeling. For any algorithm of modeling (W, denote by M · ; W, y, 6o,t(·)),

t G T, and £t0,t(·) £ Ξ (θιί , we v(; W, У, &,*(·)))

a unique (У, £t 0j t(-))~P rocess with the initial state W(£(io))· Let

be an operator which maps each measurement £(·) into the function v(; W, У, £(·))·

12

V. I. Maksimov

Definition 1.1.9. An operator D[W, is called realizing for a modeling algorithm (W, У); and a pair (w{-, W, У, £(•)),«(·, W, У, €(•))» where ξ(·) 6 Et0t$ is called a (W, У, £(·))-process. Remark 1.1.2. Further, we meet the situation when a model depends on a parameter h € (0,1): Μ = M(wο, ξ(·), v(·), h). However, we show later that the essential role in solving the problems is played by values of Μ only on elements dependent on h: wq = гуо.л, £(·) = 6ι(·)> ν(·) = vh(·). Therefore, introducing an operator M, we omit the argument h mentioning that Μ depends on h, i.e., Μ = M(wo^, 6(·)> vh(·), h). Lemma 1.1.1. For any modeling algorithm (W, У), the operator D\W, realizing this algorithm is a Volterra operator. Proof. We suppose that W is a ίο-algorithm and У is an admissible strategy. We take an arbitrary t ET, ξι(·), 6 ( 0 6 Ξ(θι# such that 6(e) = 6(e)

for

sG[t0,t}.

t»i(e)=V2(e)

for se[t0,t].

(1.1.6)

We need to show that (1.1.7)

Here vj(·) = D[W, У\,

(«,(•) = t»(·; W, У, &(•)))•

Suppose that Wj(-) = 4 ; Wo, У, (,•(•)),

3 = 1. 2.

6 = 6(ίο) = 6(ίο)

and set *&(·) = Ш м . з = i,2. Since (Wj(·), Vj(·)) is а (У, 6(0)-process with the initial state wo = VV(£o), we have wj(-) = Mt(w0, ξ}(·), vj(·)),

vj(-) = yt(6(·),

«»i(·)),

3 = 1, 2.

Hence ^(•)![ * e [0 : m - 1], бо.тЛ·) € Ξ u>(·)) € ^t0ie x Wto,0 is defined from the condition «*(•) =W(r 0. This operator is also denoted by У. A positional strategy У = (Δ, U) represents the convenient and feasible way of control of the model. At each moment τ*, on the basis of the history £t0,Ti(·) of measurement and the history Wt0,Ti(·) of motion, following rule (1.1.9), we construct a control fed onto the model on the half-interval (T»> Ti+1]· The process of forming a model control is subdivided into the finite number of steps (m — 1). So, the quadruple (M, Vl^, Ah, Uh), for each h € (0,1), defines a certain algorithm D/, in the space of measurements ξ(·) G Ξ(^(·), h), (Oh, : Ξτ Uτ) which forms the output vh(·) = Dh£(·) following the feedback principle (1.1.4), (1.1.5), (1.1.9), (1.1.10). Note that the algorithm Dh is a Volterra algorithm. We construct the regularizing family Dh, h Ε (0,1) from such algorithms; we identify each algorithm Dh with the quadruple (M, Wh·, Δ/,, Uh)· So, we consider the problem of construction of regularizing families of algorithms Dh = (M, VV^, Ah, Uh), h e (0,1) of the form (1.1.4), (1.1.5), (1.1.9), (1.1.10). The algorithm Dh (if h is fixed) has the following work scheme. Before the moment to, we choose and fix a partition A = Ah =

{Τe^Uto,o

is identified with У; the value of D on an element £(·) G Ξ T 4 ] ) >

i€[0:

m — 1].

In this case, Problem 1.1.1 is reduced to the following problem.

V. I. Maksimov

16

Problem 1.1.2. It is necessary to construct a family of positional strategies Уь such that the correspondent family (D^) of Volterra operators is regularizing. Lemma 1.1.2. If Condition 1.1.1 holds, then any positional strategy is admissible. Proof. Suppose that У = (Δ,ΖΛ) is a positional strategy and Δ is a partition of T. From the definition of У we see that it is a Volterra operator. Let t € T, wo € Wo, and £t0,t(·) € Et0lt be arbitrary. We need to show (see Definition 1.1.7) that the process (3>, £t0,t(·)) with the initial state wo exists and is unique. For this purpose, we apply the method of induction. Consider the case t Ε [ ^to,t(-)l[to,to])' m 0 ,t(·) =

(1.1.14)

6o,t(·), t*o,t(·))·

(1-1-15)

Following (1.1.3), wto,t(io) = therefore, in (1.1.14), wt0it(-)l[t0,to] replaced by wto,t(")l[io,to]· By the definition of У, this means that t w ( - ) = 3>t(ftolt(·),

w

to,t(·))·

can

be

(1-1-16)

Relations (1.1.15), (1.1.16) show that K 0 i t ( · ) , v to , t (·)) is a 0>,6 0 ,t(·))process with the initial state WQ (see Definition 1.1.6). We prove now that it is unique. Suppose that (u>to,t(·), t>t0jt(·)) and (wto,t(·), vto,t(·)) are two (;V,£t0it(-))-processes with the initial state wo- Since «t0>t(·) = X(6 0 ,t(·), üto,«(·)), we see that, by the definition of У, vt0tt{·) coincides with the right-hand side of equality (1.1.14). We show above that this equality holds true. Therefore, VtQ,t(·) = Vt0,t(')· Then wto,t(·) coincides with the right-hand side of (1.1.15), i.e., wt0,t(·) = ЩоА')· Uniqueness is proved. Now, if we assume that a (;y,£t0)t(-))-process with the initial state wo exists and is unique for each t e [t0,Ti], where i G [0 : m — 1], we can show that it exists and is unique for t € [rj, Tj+i]. We omit the detailed proof of this fact. So, the theorem is proved. •

Chapter 1. Problems of dynamical modeling in abstract systems

17

Definition 1.1.12. If Condition 1.1.1 holds, a modeling algorithm (W, У), where У = (A,W) is a positional strategy, is called positional and is denoted by (W, Δ, U).

1.2.

METHOD OF STABILIZATION OF LYAPUNOV FUNCTIONALS

We describe here sufficiently general method for constructing a regularizing family of modeling algorithms. This method consists in stabilizing an appropriate functional of Lyapunov type along the model motion. Suppose that for each /i G (0,1) we fix a functional AH :

ZT0IE χ UTO,E x WTo,0

R.

Definition 1.2.1. A family (AH) is called estimating if for any output z(·), family (wo,h) of elements from Wo, family (VH(·)) of controls and family (£Λ(·)) of measurements «-approximating z(·), from the convergence Λ(ζ(·), t»fc(·), M(w0,h, ξ Λ (·), vh(·)))

0

as

h-+Q

as

Л-Ю

(1.2.1)

it follows that the convergence (see (1.1.2)) 0M-)>

-Ю

(1.2.2)

takes place. Definition 1.2.2. A family (Wh, Уи) of modeling algorithms is A/i-stable if for any output z(-) and family (ξΛ(·)) of measurements «-approximating z(-) the convergence Λ Λ (*(·), t»(·; WA> Ун, ξ Λ (·)), «;(·; WÄJ Ун, ξh(·)))

0

as

h 0 (1.2.3)

holds. Lemma 1.2.1. Suppose that a family (AH) of functionals is estimating. Then аду (Л/,)-stable family of modeling algorithms is regularizing.

V. I. Maksimov

18

Proof. We suppose that (WH, Ун) is an arbitrary (Л^)-stable family of modeling algorithms. We take an arbitrary output z(·) and a family (ξΛ(·)) of measurements «-approximating z(·). We set vh(·) = v(-, Wh, Ун, ξh(·)), Ы-) = 4 ; Wfc) Ун, ξΗ(·)),

v>o,h = Wh(th(to)).

Then M{W0>H, £H{·), VH{·)) = WH(·)· Hence, it follows that convergence (1.2.3) (which holds since the family of modeling algorithms (Wh, Ун) is (A/,)-stable) can be written in the form (1.2.1). Since the family of functional AH is estimating, convergence (1.2.1) yields convergence (1.2.2). By the definition of the operator D\WH, Ул], we have D[Wh, «Λ(·)

= νΛ(·),

so the last convergence can be written in the form ß(D[Wh, Λ]ξ"(·), E W ) ) ) The lemma is proved.

0

as

h -> 0.

•

Further we consider, as a rule, only mean square criterions of approximation error. Suppose that the space U of controlled parameters of the dynamical system A is a uniformly convex Banach space with a norm | • the set Utoft of all controls of the system A belongs to the space L2 (T; U) (the Lebesque measure in Τ is fixed); and the criterion of approximation error has the form p{u{.), »(.)) = |«(.) - v(-)|l2(T;£/)

MO, t»(.) € Ut0,e).

(1.2.4)

Suppose also that the choice criterion has the form Ω(·)

= I · \L2(T-,U)·

Let the dynamical system A be such that for any XQ G XQ, «ι(·), Ut0,θ, A(XQ,

ui(·)) = Λ(ζο, «2(·))>

if UI(T)=U2{T)

«г(·) G

almost everywhere in T.

Therefore, taking into account (1.2.4), we identify any two controls ui(i) and ii2(t) such that щ(t) = U2{t) almost everywhere in T. Below we use the following two conditions.

Chapter 1. Problems of dynamical modeling in abstract systems

19

Condition 1.2.1. For any output z(·), the set U*{z{·)) of ω-normal controls compatible with z(·) is a singleton: UM·))

= Κ ( · ; *(·))}·

Condition 1.2.2. The set Ut0le is weakly closed and weakly compact in L 2 (T; U). We show how to construct an estimating family of functionals. Suppose that for each h Ε (0,1) a functional Лл :

Zt0!e χ Wt0,e

К

is fixed. Definition 1.2.3. A family Лд is called weakly estimating if for any output z(·), family (too,л) of elements from the family Wo, family of controls {vh(·)), family (£л(·)) of measurements «-approximating z(·), and sequence (Л*), hk 0 as к oo such that £fcfc(*(•), M(w0,hk, ξΗ), v

hk{·)

ν(·)

weakly in

«Α»(·)))->0, J)

as

к

oo,

the inclusion «(•) e U(z(·)) holds. Lemma 1.2.2. Suppose that a) Conditions 1.2.1 and 1.2.2 bold; b) a family (A^) is weakly estimating; c) Λλ(*(·), О,

w(-)) = 9(h, Äh(z(·), «;(.)), |«(-)|La(rito.w*(o) (1.2.5)

(ζ{·) e Ztθ!θ,

ν(·) Ε ut0,e,

u>(·) € Wt0i6),

20

У. I. Maksimov

where the function Φ : M+ χ Μ χ R+ χ Κ+ I is such thai for hk ->· 0 the condition $(hk, ak, bk, cfc) ->· 0 yields a k ->· 0

and

lim (6jt - c*) < 0. fc->o

Thea the family ( Λ / J is estimating. P r o o f . Assume the opposite: there exist an output z(·), a family of elements (ίϋο,/ί) from Wo, a family of controls (υ/ι(·))ι a family of measurements (£ Λ (·)) «-approximating z(·), and a sequence {hk}, hk 0 as fc —)• oo such that (*(·), t»Äfc(·), M(w0,hk,

ξ Λ *(·). VAfc(-)))

0.

However, ß{vhk (·), UM·)))

= inf{|K(·) - vhk{-)\L2{T-m I «(•) G UM·))}

=ε>0· (1.2.6)

Without loss of generality, taking into account Condition 1.2.2, we can write vh

k (·)

v(·) weakly in L2{T\U)

as

к -»· oo, υ(·) 6 Ut0ie(1.2.7)

Besides, taking into account c), we have Ahk(z(·),

M(w0 0,

as

A

^™Ι ϋ Λ»(·)Ιΐί(Γ;Ι0 — ω^(·)·

oo,

(1.2.8) (L2·9)

Hence, taking (1.2.7), (1.2.8), and b) into account, we obtain «(•) e U(z(·)).

(1.2.10)

In turn, (1.2.10) and the properties of weak limit yield the inequalities: J™>fc»(-)lL a (r;£0 > IOIl 2 (T;10 > ««>«(.)• к->oo By Condition 1.2.1, the set U*(z(·)) is a singleton:

ад·))

= {«.(•; *(•))}•

(1.2.11)

Chapter 1. Problems

of dynamical modeling

in abstract systems

21

Therefore, relations (1.2.12)-(1.2.14), by Theorem 5.12 (Gaevskii, Ch. et al., 1978), yield *(·)), Vhk(·) -> «*(·; *(•))

(1.2.12)

in L2{T;U)

as

к -»• oo.

However, (1.2.12) contradicts (1.2.6). The lemma is proved.

•

Definition 1.2.4. A family of modeling algorithms (W^, Уь.) is called (Ад)-stable if for any output z(-) and a family (£h(·)) of measurements napproximating z(·) the inequalities K ; WA> yht

ξΛ(·))|ΜΓ;

и) < Шиг{.}

+ k2(h),

(1.2.13)

ΛΛ(ζ(·), 4 ; Wh, yh, th(·))) < k3(h)

(1.2.14)

hold. The functions fci(·), МО, MO. • [0, oo)

[0, oo),

kj(h) = kj(z(.), ξΗ(·), w(-, Wh, Ун, ξΛ(·))) are such that ki(h) —>• 1,

k2(h)->0,

ks(h)

0

as

h

0.

Lemma 1.2.3. Suppose that the conditions a), b) of Lemma 1.2.2 bold. Then any (A^)-stable family of modeling algorithms is regularizing. Proof. The proof is similar to the proof of Lemma 1.2.2. We perform it for completeness of presentation. Suppose that a family of modeling algorithms (W/i, Уь) is not regularizing. This means that there exist an output z{·), a family of (£л (-^-measurements «-approximating z(-), and a sequence {ft*}, hk ->· 0 as к -)· oo such that for vhk(-)=v(-,

Whk,yhk

£4)),

u>hk(·) = Wh„,yhU thk(·)), w o , h k = m h k m inequality (1.2.6) holds. Without loss of generality, assume that convergence (1.2.7) holds. Inequalities (1.2.13), (1.2.14) yield relations (1.2.8), (1.2.9).

V. I. Maksimov

22

However, the family (Ah) is weakly estimating. Therefore, inclusion (1.2.10) holds true. Besides, (1.2.10)-(1.2.12) hold also. This contradicts (1.2.6). The lemma is proved. • Suppose, further, that for each ft e (0,1), a functional Xh : Τ χ Zt0· Ш is fixed. Definition 1.2.5. A family (Xh) is called α family of Lyapunov functionals (adjusted with a family (Лд)), if inequalities (1.2.13), (1.2.14) follow from the relation sup Xh{Ti,h, z{·), vh{·), wh{·)) < u(h), ie[0: mh] where v(h) = ι/(ζ(·), vh(·), wh(·)) -4

as

0.

Here the partition Ah is defined, following (1.1.1), as Vft(.)

= V(.; wh,

yh, ξh(•)),

u»Ä(·) = 4 ; Wfc> Ун,

th(·)).

Definition 1.2.6. A family (Wh, Ун), Ун — (Δ Λ , Uh) of positional modeling algorithms stabilizes a family (Xh) if for any h Ε (0,1), output z(·) £ Zto>e, and measurements ξ Λ (·) £ Ξίθ)β satisfying the relations К(£Л(')> z (·)) ^ ^ o n e c a n indicate a family (ψΗ,ί), к = 1 , 2 , . . . , г G [0 : тн] of nonnegative numbers h,i(Hh)> h'i *(•)» «ά(·))>

= fc(·)) is the {Wh, Ун, £Ä(-))-proces8. Lemma 1.2.4. Suppose that the conditions of Lemma 1.2.3 hold. Then алу family (VV/i, Ун), Ун = (Δ/ι, Uh) of positional modeling algorithms which stabilizes a family of Lyapunov functionals (λ/,) adjusted with a family (Λ/ι) is regularizing. Proof. Let (Wh, Ун), Ун = (Δ/ι, UH) be an arbitrary family of positional modeling algorithms which stabilizes a family (λ/J. Here AH = Кл)£о» s ( h ) = S(Ah) 0 for h -»· 0. Following Lemma 1.2.3, it suffices to show that the family (W/,, Ун) provides relations (1.2.13), (1.2.14). Fix a family of numbers {φπ,ϊ) mentioned in Definition 1.2.6. Then convergence (1.2.15) holds, and, for any t e [0 : т н ~ 1], we obtain λ Λ (τ ί+ ι ιΑ , ζ(·), νΛ(·), wh{·)) < Xh(Ti!h, ζ{·), vh(·), wh{·)) J /i

h

z

+¥'/m( ( )> \ (·), λ/»(ίο, ζ{·), Μ ' ) , гол(·)) < Ψο{h).

(1.2.17)

w

h{·)),

Summarizing all these inequalities from г = 0 to г = т н — 1 and taking into account Definition 1.2.5, we have i Afc(ri+ilh, «(•), t»A(·), wh(·)) hj(S(h), h\ *(·), u/fc(·)) j=о *(•)> Μ'), Wfc(·))· Since, following (1.2.15), (1.2.16), the right-hand side of this inequality tends to zero for h -» 0+, we obtain (1.2.13), (1.2.14). The lemma is proved. • It is easily seen that the following theorem is true. Theorem 1.2.1. Suppose a) a family (WA, Ун) is (Ah)-stable; b) Conditions 1.2.1 and 1.2.2 hold; c) the conditions АнМ)> 4 ; Wfcfc> yhk,thk{·))) Vk(-) = «(•; whk, yhk^hk(·))

ο,

«(•) hk -4 0

ξΗ" e ξ ( * ( · ) , hk), ^аЫу in L2(T; U), as

ft

->· oo

yieid inclusion (1.2.10). Then the family (WA, ^/i) is regularizing.

24

V.J. Maksimov

Remark 1.2.1. We may not require weak compactness of the set Ut0,e if for modeling algorithms (W/j, Ун) the set υ(·; Wh, Уь, (h(·)) is bounded in L2{T; U) uniformly with respect to all x(-) G Χ^,θ, ( h € Ξ(ζ(·),/ι). In this connection, Lemma 3.2.4 and Theorem 1.2.1 remain valid.

1.3.

RECONSTRUCTION OF DISTRIBUTED CONTROLS IN ABSTRACT SYSTEMS. THE CASE OF INSTANTANEOUS RESTRICTIONS IMPOSED ON CONTROLS

There are two difficulties when realizing the approach described above: the first one is to choose a model Μ with appropriate properties; the second one is to construct modeling algorithms (Wh, Δ/ι, Uh)· It is essential to take Μ as the copy of real system, in this case Μ is described by the same operator A as the given dynamical system. This cannot be done always. However, under certain conditions, which we formulate below, we can set Μ = A and show the explicit form of РАМ (Wh, Δ/ι, Uh) forming a regularizing family FSDAM. Now, we describe these conditions and construct corresponding РАМ. 1. The method of smoothing functional Suppose that U is a uniformly convex Banach space; X and Ξ axe Banach spaces with norms | · and | · |ξ respectively; X is imbedded into Ξ continuously and densely; Χ Ε X and Xq С X are certain fixed sets; the control set Utofi has the form Uto,e = {«(•) € L 2 (T; U) : u(t) € Ρ

for a. a.

t G T}.

(1.3.1)

Let the criterion of error approximation be as in (1.2.4) and the choice criterion be in the form ω(·) = | • \ьг(т-,и)· Hereinafter, Ρ С U is a given convex, bounded, and closed set (the set of control resources). Note that continuity of imbedding of the space X into the space Ξ means that there exists Co G (0, oo) such that N =

< CQ\X\X

for all

χ Ε X.

Suppose that there exists a mapping such that for any moment f* G T, state χ* G X, and control u(·) G Ut0ig there exists a function я(·; U, x*, u(·)) : with the properties:

[ί*,θ]4·Χ

Chapter 1. Problems of dynamical modeling in abstract systems a) x(U;

25

U, ζ», u(·)) = x*;

b) if t £ [i*, 0], tti(·), U2(·) £ Ut,,e are such that UI(T) = иг(т) for a.a. r G [ί„ i], then ж(т; U, x„ u\(•)) = X{T\ U, ж*, ii2(·))

for all

r G (U, t].

The property b) allows us to introduce a mapping which associates each U e τ , χ* e x , t e [u, 0], ut.,t(·) £ Uu,t with x(-t u, x„ ut,lt(·)) = ®t.,t(·; U, X*, «(·)), where u(·) £ U^e is such that и{т) = гtt,,t(r)

for a.a.

t £ [£», t].

In this section, we take Χ*,*

ίο,

=

"(О)

:

χ0

6 Xo, «(•) £ Utof}}

and a controlled dynamical system of the form A(x0, u{·)) = x(·; i0) ®o, «(•))· A motion (of the system A) in an interval i*] С Τ corresponding to an initial position (i„, ж»), χ* £ X and a control (•) G is a function t x(t) = x(t; t„ x„ ut„r(·)) : [U, t*] X. Assume that the function x(-) satisfies the following condition. Condition 1.3.1. a) (the smoothness property) For any t„, t* G Τ, t* < t*, х, £ X, and u(·) £ Ut,the function z(·; f„, x», u(·)) is continuous. b) (the semigroup property) For any £ T, t* £ (f„, 0], xo G Xo, u to,t.(·) € U u, u . - £ U ,f, and t £ [t„ t*] the equality to> t it u x{t; t0, xo, ut0,t*(·)) = x{t] t*, x*, ut.,t*{·)) where x t = z(i*; i 0 , so, «t0lt*(·)),

( holds.

uto,t*(·)

for a.a. τ £ [ίο,ί*],

iti.,t*(·) for a.a. τ G [*•, 0 such that r{x + y) < K(r(x) + r(y)) for each x,y 6 I « = I i U X2, where X\ = { x G X :

χ = χι — Х2, Xj = x(t) to, Xj, it(·)),

j = 1,2,

x*j e l o , t e τ , Uj{·) 6 Utofi }, X2 = { x € X ·• 3xi 6 XQ,X2 £ X : |®ι - X2IX < 1 ,r{x - X2) < 1 } · Suppose also that a functional L(·, •):

ΞχΡ-)Κ

satisfies the following condition: Condition 1.3.2. a) For any t,, t* € T, t, < t*, χι, X2 G X, щ{·), «2(·) G Uu,t*, the functions τ

L(®i(r) - x 2 (r),

U i (r))

: [t., Γ]

R,

where j = 1,2; ζ;·(·) = χ{·\ t* 5 Xj, Uj (·)), are integrable and the inequality Φι(ί*)

- X2(t*)) < r{Xl(U) - X2{Q) + UI(R))

- L(XI(T)

F

Jt.

{L{X\{T)

- Χ2(Τ),

holds. b) L(x,u) < с|ж|н for any и Ε Ρ (с = const G (0,00));

-

U2(T))}(1T

X2{T),

Chapter 1. Problems of dynamical modeling in abstract systems

27

c) if x(·) Ε C(T; X), t Ε Τ, ιcj(·), u(-) Ε Ut0)t and Uj{·) -4 u{·) weakly in L2([to,t]·, U) as j oo, then lim f {L(x(t), i ^ o o j t0

иЛт)) - L(x(t), w(r))}dr = 0;

d) for any χ Ε Ξ the functional и —> L(x, и) is weakly lower semicontinuous; e) the functional χ ->• L(x, u) : Ξ —>• R is linear for any и Ε P. We pass now to solving the problem. Suppose that the information operator С = I (the identity operator), Ξο = Ξ, the measurement set Ξίθι# is a subset of the set of bounded functions ξ(·) : Τ —»· Ξ, [CQl\in(to)-xo\x,i{i η - Tith,

W = X,

Wo-

Xo,

Wtoß

= 0, - XtQft.

We take Μ = Aas the model; i.e., we take the copy of the controlled dynamical system. A motion of the model in an interval [• 0, 03(/ι) + ι / ( / ι ) ) α _ 1 ( Λ ) a s

h ->· 0.

(1.3.2) (1.3.3) (1.3.4)

28

V. I. Maksimov

T h e function α ( · ) plays the role of the regularizator (following Tikhonov, A . N . ) . Condition 1.3.1, c) and convergence · 0 as h ->• 0 yield the relation ί(ΔΛ)(1 + 0 ( Δ Λ ) ) - Κ )

as

Л-Ю.

(1.3.5)

Now, we construct the positional strategy ( Δ ^ , Uh), h G ( 0 , 1 ) (see Definition 1.1.11) setting the values U h ( n , & , , „ ( • ) , w t 0 > T i (·)) = < T j + 1 ( · )

(1.3.6)

of the mapping Uh (here and below, we set, for brevity, т ^ = ц ) following the rule

a-a·

for

* € [η, r i + i ) ,

(1.3.7)

where ν? = a r g m i n { 7 ( a , v, Si) : υ e P}, / ( a , v, Si) = L ( s i , ν) + a ( h ) \ v \ u ,

(1.3.8)

ξ(η).

Si = Ц т * ) -

W e introduce the set W ^ of io _ algorithms as follows: Wft(e(to))=tüo,

(1.3.9)

where G Β0(ξ(ί0)) = {w G X 0 : Γ ( ξ ( ί 0 ) - ю ) < u{h)}, if г«о = год1^ otherwise; w^ that

Д , (£(*>)) φ 0,

is a n arbitrary element from the set XQ. Suppose

Xh(t, * ( · ) , »(•), w ( - ) ) = а - Ч Л М Ц * ) - * ( * ) ) + Al®(r)&-Mr;

x(-))| 2 tr}dT,

Jto

Ah(x(·),

v(-), «,(•)) = a - H h )

+ Xfc(®(·), w ( · ) ) =

Jto

sup r(w(r) те[го,в]

(1.3.10)

- х(т))

(1.3.11)

[*{Ш\Ь-Ыт1

sup T > ( i ) - s ( i ) ) . te[t0,9]

(1-3.12)

Chapter 1. Problems of dynamical modeling in abstract systems

29

Here «•(•J ®(·)) = argmin{|?;(-)|L2(T; υ) • υ(·) G

U(x(·))

tf(®(·))},

i 0 , ®o, " ( О ) Vi G T}

= (u(-) 6 Ut0te : x(t) = x(t;

(1.3.13)

is the set of controls u(·) G Ut0>e compatible with the output x(·). Theorem 1.3.1. Suppose that Conditions 1.2.1, 1.3.1, 1.3.2 hold. Then 1) the family of functionals (Ah) (1.3.12) is weakly estimating; 2) the family of functionals (Лд) (1.3.11) is estimating; 3) the family (Лд) of the form (1.3.10) is the family of Lyapunov functionals (adjusted with the family (Ah) (1.3.12)); 4) the family of positional modeling algorithms (Wh, Ah, Щ) deßned as in (1.3.6)-(1.3.9) stabilizes the family of Lyapunov functionals (Xh) (1.3.10). Proof. 1. Suppose that the family (Ah) is not weakly estimating, i.e., there exist an output χ(·) — x(·] TO, XQ, U(·)), a family of elements (wo,h) from Wo, a family (υ/ι(·)), Vh(·) G Ut0je of controls, a family of measurements (£ Λ (·)) «-approximating x(·), and a sequence {hk}, hk ->• 0 as к oo such that A h k (x(-),w h k (•)) wo,hk

0, vhk(·) x(to)

in

w(·) weakly in L2(T; X

as

к

U),

oo,

where ™hk(·) =M(wQihk,

vhk(·))

=ω(·;

«(•) ί

U(x(·)).

i0, wQthk,

However,

This means that for some i« G (ί, Θ] we have \x(U)-w(Q\x >0.

vhk(·)).

(1.3.14)

V. I. Maksimov

Here ж(·) = ж(·; ί0, so, «(•)),

ω(·) =υι(·;

tQ, χ0,

ν(·)).

Therefore, taking into account the property a) of the functional r(·), we obtain the inequality (1.3.15)

r{x(U) - w{U)) > ε. Further, by the property b) of the functional r(-), we have r(x(U)

- w(Q)

< K{rhk(U)

ri,k{t) = r(x{t) - whk(t)),

+

r2,k(U)},

r2,k(t) = r(whk{t) - w(t)).

Taking into account Condition 1.3.2, a), b), e), we see that г2>к(**) < г Ы , н к - x{h))

(1.3.16)

+ J^ [L{whk (r) - w(r), vhk (r)) v(r))}dr

-L{whk(T)-w{r), < r2,k(to) + J^

{ L{whk{r) - x(t), - L(whk{r)

vhk{r))

- X(T), V(T))

+ L(x(t) — w(t),

vhk(r))

— L(x(t) — w(t), v(t)) I dr и < r2,k(to) + 2с [ K f c ( r ) - x ( r ) | s d r Jt D + /

Щх{т) - w{r),vhk (r)) - L{x(t) - w(t), υ(τ))} dr.

Jto

The number с is defined in Condition 1.3.2 b). This, (1.3.1), (1.3.14), and Condition 1.3.2 c) yield the relation lim {rltk{U)+r2,k{U)} k-+oo

= 0.

This relation in virtue of (1.3.16) contradicts (1.3.15). Hence, it follows that the family (Ah) is weakly estimating.

Chapter 1. Problems of dynamical modeling in abstract systems 2. We consider the function Φ : R+ χ Μ χ Μ+ χ R+

31

Μ,

αα 1 (h) + Ь — с, if b, с < d,

a, 6, c) =

+00,

otherwise,

where d = sup{|u(-)U 2 (r ; u) • «(·) e Ut0,e}· For such choice of Φ the functional Δ^ of the form (1.3.11) is taken by formula (1.2.5). Therefore, if ${hk, aki h, ck) ->· 0, then ak

0,

lim (bk - ck) < 0. fe-юо

So, all conditions of Lemma 1.2.2 hold. Therefore, the family of functionals Aft is estimating. 3. Now we show that the family (λ/J of the form (1.3.10) is the family of Lyapunov functionals (adjusted with (Λ^)). Suppose that the inequalities sup

Xh(n,h, *(•). vh(·), wh(·)) < u(h),

(1.3.17)

ie[0: τη/,]

where v{h) = v(z(·), vh{·), wh(·)) vh(-)=v(-,wh,

h

yh^ (·)),

0

as

h

wfc(-) = 4 ;

0, УА> £*(·)) (1.3.18)

hold. Since the functional r(-) is nonnegative, (1.3.10), (1.3.17), and (1.3.18) imply (1.2.13) for h(h) = 1,

k2{h) = ιM2{h).

Inequality (1.3.17) yields r{wh{Tiih) - x(ri>h)) < v,{h) = a(h)u(h) + 2a{h)($ г 6 [0 : m h ], where d{P) = sup{Hi/ : и e P}.

t0) *(·)» «**(')» Whk{·)) +

*(").

w

hk(·))·

Taking into account (1.3.9), the property b) of the functional z(·), and the definition of κ(·), ι/(·), we obtain Xhk(t0, x(-), « 4 ) ,

^(·))0.

As the model M, we take "the extended copy" of the system A, i.e., the mapping M(w0,

£(·), • (w^(·),

wW{t) e к

«(•))=*(·;

Vi g τ,

w^(·)}, wo = (uft

?\ «(Ο),

wW(t)=

0),

[г\у(т)\Ь0т. Jto

Definition 1 . 3 . 1 . For a point w= а (го, ξ, и)-concomitant

щ(2')бХхМ,

ξ€Χ,

point is a pair ( г ^ , ζ

6

= a r g m i n { | z - £ | ; r : r{z - tu (1) ) < и2, z^ = arg min{z : r{z^

- w^) + \z -

и > 0, X х Κ such that ζ G Χ},

w(2)|2

< ν2,

ζ

еЩ.

If r(·) = I · then, as it is easily seen, we have г(1)

ί

if

ξ, (1)

\ wW + ι/(ξ - U> )/|£ - IÜ(1)\x

eeS^1);

ν),

otherwise,

Here S(wW·, i/) = {x € X '. Ιω*1* - χ\χ < v}. The positional strategy Ун = (Δ&, Uh) is defined according to (1.3.6), (1.3.7) as follows: t£ = a r g m i n { I ^ 1 ' - ^ 1 ' ,

u) + 2{wW - z^)\u\2v : и G Ρ}.

(1.3.25)

Chapter 1. Problems of dynamical modeling in abstract systems

39

Here ( z ^ , z^) is a (w, ξ, щ(т^) )-concomitant point. The family of ίο-algorithms (W/ι) is defined following rule (1.3.9), where we set WO =

0),

u»S1} e Βο(ξο) = {u>(1) € X0 : \ξο -

< o, 5h},

if B0 φ 0, and wy( i ) =X о otherwise. Theorem 1.3.3. Let the conditions of Theorem 1.3.1 hold. Then the family of positional modeling algorithms (Wh, Δ^, Uh) of the form (1.3.6), (1.3.7), (1.3.9), (1.3.25) is regularizing. Before we prove the theorem, establish the lemma. Denote by x^; ν) the set of points (zW, z^>) e l x l satisfying the inequality ф(1)_ж(1))+ m i n \ZW-X\*29)

It is evident (see (1.3.27), (1.3.28)) that x W { t ) < x W ( t ) ,

t E [ U , t * } .

In this connection, for proving inclusion (1.3.26) it suffices to show that u ( t ) =

u ( y W ( t ) -

®W(t)) + Ιί/(2)(ί) - x (2) (i)| 2 < и?(*)•

(1.3.30)

Taking into account Condition 1.3.2, a), we have u(t) < r(yW - xW) + A i ( j / ( 1 ) ( r ) - X (1) (r), tx,(r)) Jt.

-L(yW{T)-xW(T), + b ( 2 ) - ® o | 2 + /i(i., t) + I2(U, t),

^(r))}dr (1.3.31)

Chapter 1. Problems of dynamical modeling in abstract systems

41

where h(U, t) = 2 (y - s 0 ) A M r ) & - b(T)|2D}dr, Jt.

Ь)=([\ыг)\Ь-\у(г)\Ь}йт)2.

h(U, Jt.

Using с) and the definition of the number xo we deduce that r („(i)

_ a;(i)) + |y(2) _ Жо|2 < v 2 ( L 3 32) Let us estimate the second term in the right-hand side of inequality (1.3.31). Taking into account Condition 1.3.2 b) and inequality (1.3.29), we obtain A L ( y V ( T ) - x W ( T ) , u,(T))-L(yW(T)-*W(T), v(r))}dr Jt. < A L ( y M ( r ) - z U , u>{r))-L{yW Jt. +(t-U)c{

-zW,

(1.3.33)

v(r))}dr

max | j / ( 1 ) ( 0 - y ( 1 ) U + max ^ ( O - i ^ U + u(h)}. fe[t»,t] ,t]

Besides, we have h(U, t) (•) 6 Ξ(χ(·), hj) such that hj -> 0 as j oo and Ι^(·)-^(·)ΐΜΓ;Ε/)>εο>ο,

«,(·) = «.(·; *(·))·

(1-3-36)

Taking into account Lemma 1.3.2, inequalities (1.3.22), (1.3.23), and the inequality r(zo < nh],

42

V". I.

Maksimov

by the induction method, we establish the inclusion x (2) (i),

jA'((t) w w

= h

(1.3.37)

t ET.

u>M2)(t)) is the phase trajectory of the model,

(whi^(t),

^ ( t ) = x ( f , ίο, Й',

W ) ( t ) =

for

vhj(t))

«Ч)),

f \vhj(τ)\υάτ, Jto

xW(t)

=

f Jto

|u*(r)|ydr.

We set Vj = sup ν ε . (t). t€T

Then (see (1.3.24)) Uj -> 0

as

j

oo.

(1.3.38)

Inclusion (1.3.37) for h < hj means that r{wh^(t)

- x(t))

+

min

®(t)

I w h № ( t ) - x\2