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K>0 , а>0 К< 0,а>0 а) ->0 + К>0, а< 0 / /(- 1 ,0 ) а > -» + оо g o -»- оо I 1 T VOL.II \ LECTURES PRESENTED N AT AN v j l cj -»0- INTERNATIONAL SEMINAR COURSE / а<0 TRIESTE, 11 SEPTEMBER - 29 NOVEMBER 1974 ORGANIZED BY THE INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS TRIESTE O J - . 0 + - INTERNATIONAL ATOMIC ENERGY AGENCY, VIENNA, 1 976 • r r * s CONTROL THEORY AND TOPICS IN FUNCTIONAL ANALYSIS Vol.II INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS, TRIESTE CONTROL THEORY AND TOPICS IN FUNCTIONAL ANALYSIS LECTURES PRESENTED AT AN INTERNATIONAL SEMINAR COURSE AT TRIESTE FROM 11 SEPTEMBER TO 29 NOVEMBER 1974 ORGANIZED BY THE INTERNATIONAL. CENTRE FOR THEORETICAL PHYSICS, TRIESTE In three volumes VOL. II INTERNATIONAL ATOMIC ENERGY AGENCY VIENNA, 1976 THE INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS (ICTP) in Trieste was established by the International Atomic Energy Agency (IA E A ) in 1964 under an agreement with the Italian Government, and with the assistance of the City and University of Trieste. The IAEA and the United Nations Educational, Scientific and Cultural Organization (UNESCO) subsequently agreed to operate the Centre jointly from 1 January 1970. Member States of both organizations participate in the work of the Centre, the main purpose of which is to foster, through training and research, the advancement of theoretical physics, with special regard to the needs of developing countries. CONTROL THEORY AND TOPICS IN FUNCTIONAL ANALYSIS IAEA, VIENNA, 1976 STI / PU B /415 ISBN 9 2 -0 - 1 3 0 1 7 6 - 6 Printed by the IAEA in Austria May 1976 FOREWORD The International Centre for Theoretical Physics has maintained an interdisciplinary character in its research and training programmes in different branches of theoretical physics and related applied mathematics. In pursuance of this objective, the Centre has - since 1964 — organized extended research courses in various disciplines; most of the Proceedings of these courses have been published by the International Atomic Energy Agency. The present three volumes record the Proceedings of the 1974 Autumn Course on Control Theory and Topics in Functional Analysis held from 11 September to 29 November 1974. The first volume consists of fundamental courses on differential systems, functional analysis and optimization in theory and applications; the second contains lectures on control theory and optimal control of ordinary differential systems; the third volume deals with infinite dimensional (hereditary, stochastic and partial differential) systems. The programme of lectures was organized by Professors R. Conti (Florence, Italy), L. Markus (Warwick, United Kingdom) and C. Olech (Warsaw, Poland). Abdus Salam CONTENTS OF VOL.II Time-optimal feedback control for linear systems (IAEA-SMR-17/46) ................................... 1 S. MiricS Applications of functional analysis to optimal control problems (IAEA-SM R-17/47)........ 39 K. M izukami Non-linear functional analysis and applications to optimal control theory (IA E A -S M R -17 /3 0 )............................................................................................................................ 75 C. Vârsan Some topics in the mathematical theory of optimal control (IAEA-SM R-17/46)................. 157 T. Zolezzi An application of Pontrjagin’s principle to the study of the optimal growth of population (IA E A -S M R -17 /4 9 ).................................................................................................. 189 Vera de Spinadel Functional analysis in the study of differential and integral equations (IA E A -S M R -17/5 0 ).............................................................................................................................. 201 G.R. Sell Spectral theories for linear differential equations (1AEA-SM R-17/51)..................................... 219 G.R. Sell Some examples of dynamical systems (IA E A -S M R -17/66) ......................................................... 227 S. Shahshahani Realization theory of linear dynamical systems (IAEA-SM R-17 /5 3 ) ........................................ 235 R E. Kalman Basic equation of input-output models and some related topics (IA EA -SM R -17 /8 1 ) ........... 257 M. Ribaric Support functions and the integration of set-valued mappings (IA E A -S M R -17 /5 9 ) ............. 281 G.S. Goodman Topics in set-valued integration (IA E A -S M R -17/60) ..................................................................... 297 R. Pallu de la Barrière IAEA-SMR-17/46 TIME-OPTIMAL FEEDBACK CONTROL FOR LINEAR SYSTEMS S. MIRICA Department of Mathematics, University of Bucharest, Bucharest, Romania Abstract TIME-OPTIMAL FEEDBACK CONTROL FOR LINEAR SYSTEMS. The paper dèals with the results of qualitative investigations of the time-optim al feedback control for linear systems with constant coefficients. In the first section, after some definitions and notations, two examples are given and it is shown that even the time-optim al control problem for linear systems with constant coefficients which looked like "completely solved" requires a further qualitative investigation of the stability to "permanent perturbations" of optimal feedback control. In the second section some basic results of the linear time-optimal control problem are reviewed. The third section deals with the definition of Boltyanskii's "regular synthesis" and its connection to Filippov’s theory of right-hand side discontinuous differential equations. In the fourth section a theorem is proved concerning the stability to perturbations of time-optimal feedback control for linear systems with scalar control. In the last two sections it is proved that, if the matrix which defines the system has only real eigenvalues or is three-dimensional, the time-optim al feedback control defines a regular synthesis and therefore is stable to perturbations. 1. THE LINEAR TIME-OPTIMAL CONTROL PROBLEM. NOTATIONS, DEFINITIONS AND EXAMPLES We con sid er an n X n real m atrix A € L(Rn, R n), an n X p real m atrix В e L(RP, Rn) and a closed, convex, p-dimensional polyhedron U С Rp such that 0 £ U, but 0 is not a vertex of U. The elements A, B, U define the linear parametrized differential system ^ = Ax + Bu, x G Rn, ueU (1.1) dt Rn is called the phase space and U the control space. An admissible control with respect to the state x' 6 Rn is a piecewise continuous map u:[t0, t j -» U such that the solution cpu(.; t0, XgJ^tg, t1J - Rn of the "controlled" differential system H v — = Ax + Bu(t), x(t0) = x 0 (1.2) reaches the origin 0 G Rn for the first time at the moment tj> tQ (that is, <pu(t;t0, x0) € Rrt\ {0 } for t e [t0, tx) and 9 u(t1; t0, x0) = 0). We say that the control u(.) steers the state x 0 to the origin in the time ^ - tQ;. the solution cpu(.; t0, x 0):[t0, tjj -► Rn is called an admissible trajectory and (u(.), cpu(.; t0, tj)) an admissible pair with respect to the state x 0 € Rn. 1 2 MIRICA We recall that a map u(.): [t0, tj] -*• U is said to be piecewise continuous if there exist f о = To < Ti < ••• < 7k < Tk + l = * 1 suc^ the restriction maps u| (tí , t 1 + 1 ) , i = 0, 1 , ... к are continuous and one-sided limits exist at the points t0, Tj, ... тк + 1 . Since the values of u(.) at the discontinuity points ... тк do not count in the differential equation (1 . 2 ), we may take u(.) to be continuous to the right-hand at each point: lim u(t) = u(s) as s - t and t < s, and continuous at the ends of the interval [t0, tj]. If we denote by® ' the set of all admissible controls with respect to the state x € Rn, then the linear time-optimal control problem may be stated as follows: given A G L(Rn, Rn), B G L(RP, Rn) U С RP a p-dimensional closed, convex polyhedron (such that 0 G U, but 0 is not a vertex of U) which define the system (1 . 1 ); find an admissible control ux(.) G<% for each point x e Rn for which is not empty, such that flY(.) steers the point x to the~ origin 0 G Rn in minimal time, among all the admissible controls u„(.) G í/X. We say that <ïx(.) is an optimal control with respect to x and the corres ponding solution cpïï(.; t0, x0):lt0, txJ -► Rn of (1.2) is called an optimal trajectory of x. Therefore, the linear time-optimal control problem is a standard optimal control problem where the "terminal manifold" or the "target" is 't, the origin {0} and the optimality criterion is min( j dt). So we can apply the general results of optimal control theory to this problem, namely the theorems of existence and maximum principle. Let us denote by G С Rn the set of those point x G Rn for which the set ^ of the admissible controls with respect to x is not empty. We call G the controllability set of the system. Theorem 1.1 (existence of optimal controls [2,6,12,20]. For any state x e G there exists an optimal control ux: [0,T(x)]-> U which steers the point x to the origin 0 within the minimal time T(x). Definition 1.2 (the Maximum Principle [2]) Let H : Rn X Rn X U -> R be a map defined by H(X, x, u) = <X, Ax + Bu> (1.3) and let us consider the differential system (1.4) where A* is the transposed of the matrix A. We say that the admissible control u(.) G <8^, u(.):[t0, t j -* U satisfies the maximum principle; if there exists a non-trivial solution X:[t0, txJ -* Rn of (1.4) such that: H(Mt), q>u(t; t0, x), u(t)) = max H(X(t), q>u(t; t0, x), u) (1.5) u e U HfXitj), 9 u(tj; tQ, x), u(tj)) ë 0 (1.6) where фи(.; t0, x ^ :^ ,^ ] — Rn is the solution through x of (1.2).