Impulsive Control in Continuous and Discrete-Continuous Systems

453 Pages·2003·10.9 MB·English

by  Boris M. Miller, Evgeny Ya. Rubinovich (auth.)

Impulsive Control in
Continuous and Discrete
Continuous Systems
Illlpulsive Control in
Continuous and Discrete
Continuous Systems
Boris M. Miller
Institute for Information Transmission Problems
Moscow, Russia
and
Evgeny Ya. Rubinovich
Institute of Control Sciences
Moscow, Russia
Springer Science+Business Media, LLC
ISBN 978-1-4613-4921-1 ISBN 978-1-4615-0095-7 (eBook)
DOI 10.1007/978-1-4615-0095-7
©2003 Springer Science+Business Media New York
Originally published by Kluwer Academic / Plenum Publishers in 2003
Softcover reprint of the hardcover lst edition 2003
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To Olga and Irina
Preface
Dynamic systems with discrete-continuous properties (discrete-continuous
systems (DeS) or hybrid systems) arise in various areas of the applied sci
ences such as: mechanics, economics, telecommunications e.a. This mono
graph considers the optimal control problems for the class of dynamic sys
tems with controls causing the abrupt changes in their states. In other
words we deal with the dynamic systems admitting impulsive controls with
discontinuous trajectories. The peculiarity of the optimal control problems
for such systems is that the instants and magnitude of the impulsive actions
are not specified beforehand and must be defined by the optimization prob
lem solution. Since these systems admit the discontinuous paths they can
not be described by ordinary differential equations. Therefore, it is natural
to use a more general type of equations, namely, the nonlinear differential
equations with measures. This type of equations provides a universal tool
for the description of various modes in DeS. However, the optimal con
trol problems for systems governed by nonlinear differential equations with
measures appear to be more sophisticated than the traditional ones, and
this circumstance leads to the necessity of developing the novel methods in
optimal control problems as well as in classical calculus-of-variations.
This extension of the classical methodology to a new' class of systems
had been successfully developed in recent years. The powerful method
which can be referred to as the method of discontinuous time-change, was
developed for various control problems in the area of DeS. By using the
special transformation of the time scale, this method makes it possible to
reduce a complicated optimal control problem with impUlsive controls and
discontinuous trajectories to a standard one, which can be formulated for
some auxiliary system described by ordinary differential equations with
bounded controls.
In this monograph we consider various aspects of this discontinuous
time-change method applied to optimal control problems in DeS, and
namely:
• representations of discontinuous paths by making use of the solutions
of some auxiliary dynamic system with bounded controls,
• derivation of dynamic equations as differential equations with mea
sures,
• proof of the existence theorems in optimal control problems,
• and finally, derivation of necessary and sufficient optimality condi
tions.
vii
viii
We also consider the application of this method to the problems of obser
vations control in the stochastic DCS.
The monograph can be used as a textbook for those attending the
courses in the Optimal Control Theory for Hybrid Systems, furthermore, it
may prove useful not only to researchers, it may come in useful to practition
ers in areas such as Control Engineering, Operation Research and Applied
Control Science. No particular background is required, however, some ba
sic notions from the Theory of Ordinary Differential Equations, Theory of
Optimal Control, and introductory notions from Real Analysis and some
Functional Analysis would be useful. Moreover all necessary notions are
given within the special mathematical appendix.
The authors express their gratitude to Professors Alexandre Sesekin,
Stanislave Zavalishin (Ekaterinburg, Russia), Vladimir Dykhta (Irkutsk,
Russia),
Franco Rampazzo (Padova, Italy), Joseph Bentsman (Urbana-Champaign,
USA), Prof. Bernard Brogliato (Grenoble, France), Prof. Fernando Lobo
Pereira (Porto, Portugal) for their stimulating discussions on the subject; to
Prof. Pierre Bertrand (Gif-sur-Yvette, France) and Francois Dufour (Bor
deaux, France) and to Prof. Andrey Savkin (Sydney, Australia) for the
opportunity to continue this work in France and in Australia.
We are indebted to Prof. George A. Anastassiou (Memphis, USA) and
to Prof. V. Lakshmikantham (Melbourne, Florida, USA) for their support
in the course of this monograph preparation and its consequent submission.
We gratefully acknowledge the valuable services of INTAS, NSF of USA,
Russian RBRF, CNRS France and the Australian Research Council which
had been supporting this research at different stages in the course of some
years.
We convey our thanks to the the publishers for the fruitful collaboration
and we particularly extend our thanks to our Editor Mrs. Ana Bozicevic.
We are indebted to our wives Olga and Irina, and our families for their
inexhaustible patience and permanent support.
Bordeaux, October 2001 Boris M. Miller
Sydney, October 2001 Evgeny Ya. Rubinovich
Contents
1 Introduction 1
1.1 Concept of discrete-continuous (hybrid) system and some
typical problems . . . . . . . . . . . . . . . . . . . . . . ., 1
1.1.1 Absence of the optimal control . . . . . . . . . . .. 3
1.1.2 Linear and nonlinear differential equations with the
measure-type controls . . . . . . . . . . . . . . . .. 8
1.1.3 Approximation of the impulsive controls. . . . . .. 14
1.1.4 Instability of DCS with respect to small variation of
impulsive controls ............... 19
1.2 Robust and non-robust discrete-continuous systems. 21
1.2.1 Robust discrete-continuous systems. . . 21
1.2.2 Non-robust discrete-continuous systems 25
1.2.3 Method of discontinuous-time-change 32
1.3 The structure of the book . . . . . . . . . . . . 36
2 Discrete-continuous systems with impulse control 39
2.1 Definition of impulsive control and the system solution. 39
2.2 Stability (robustness) conditions for systems with impulse
control. . . . . . . . . . . . . . . . . . . 45
2.2.1 Necessary stability conditions . . . . . . . . . . . .. 45
2.2.2 Sufficient conditions of stability . . . . . . . . . . .. 71
2.3 Generalized solutions of systems with impulse control and
their representations . . . . . . . . . . . . . . . . . . . . .. 79
2.3.1 Definition of generalized solution and its representa-
tion by change of variables .............. 79
2.3.2 Discontinuous time change and representation of gen-
eralized solution .................... 85
2.3.3 Representation of generalized solutions by differential
equations with a measure . . . . . . . . . . . . . .. 94
ix
x Contents
3 Optimal impulse control problem with restricted number
of impulses 103
3.1 Introduction and the problem statement . 103
3.2 Auxiliary optimal control problem . . . . 105
3.2.1 Reduction to an auxiliary problem 105
3.2.2 Equivalence of the primary and auxiliary problems 108
3.3 Necessary and sufficient optimality conditions. . . . . .. 115
3.3.1 The necessary optimality conditions in auxiliary prob-
lem ........................... 115
3.3.2 The necessary optimality conditions in the primary
problem . . . . . . . . . . . . . 120
3.3.3 Sufficient optimality conditions . . . . . . . . . . .. 126
4 Representation of generalized solutions via differential equa
tions with measures 137
4.1 Generalized solutions of nonlinear differential equations .. 137
4.1.1 Definition of generalized solutions and their represen-
tation via discontinuous time change ........ 139
4.1.2 Differential inclusions for generalized solutions of non
linear differential equations . . . . . . . . . . . . .. 150
4.1.3 Representation of generalized solutions by differential
equations with a measure . . . . . . . . . . . . . .. 161
4.2 Generalized solutions of differential equations with affine
dependence on unbounded controls . . . . . . . . . . . . .. 167
4.2.1 Differential equations with affine dependence on un
bounded controls . . . . . . . . . . . . . . . . . . .. 168
4.2.2 Differential equations with the Lie algebra structure 176
5 Optimal control problems within the class of generalized
solutions 193
5.1 Statement of the optimal control problems with phase con-
straints ..................... 193
5.2 Existence of the optimal generalized solution 199
5.2.1 Existence of the weak solution .... 199
5.2.2 Auxiliary control problem. . . . . . . 202
5.2.3 Auxiliary control problem for weak solution 206
5.3 Optimal control problems for DOS with ordinary and im-
pulse controls . . . . . . . . . . . . . . . . . . . . . . . .. 208
5.4 Optimal generalized solutions in nonlinear hybrid systems 222
5.4.1 Hybrid systems with unilateral constraints. 222
5.4.2 Generalized solution of hybrid system . . . . . .. 229
Contents xi
5.4.3 Existence of the generalized solution in the optimal
control problem for hybrid systems . . . . . . . . .. 233
6 Optimality conditions in control problems within the class
of generalized solutions 237
6.1 Introduction.................... 237
6.2 Generalized maximum principle ........ 239
6.2.1 General problem with phase constraints 239
6.2.2 Necessary optimality condition in problems with phase
constraints .................. 245
6.3 Applications of generalized maximum principle . . . . . 261
6.3.1 Systems, satisfying the stability conditions 261
6.3.2 Necessary conditions for weak optimal solutions. 276
6.3.3 Necessary optimality conditions in hybrid systems 287
6.4 Generalized maximum principle in linear-convex problems 297
6.4.1 Necessary and sufficient optimality conditions in con-
vex problems . . . . . . . . . . . . . . . . . . 297
6.4.2 Optimal control problems for linear systems . . . .. 312
7 Observation control problems in discrete-continuous stochas
tic systems 325
7.1 Statement of the problem . . . . . . . . . . . . . . . . . .. 325
7.2 Generalized solutions in observation control problems 329
7.2.1 Representation of generalized solutions in observation
control problems . . . . . . . . . . . . . . . . . . .. 330
7.2.2 Solution of observation control problem via discon-
tinuous time transformation method . . . . 337
7.3 Convex properties of observation control problems 338
7.4 Examples of observation control problems . . . . . 348
7.4.1 Systems with deterministic dynamic . . . . 349
7.4.2 Observation control problems with time and energy
constraints .................. 351
7.4.3 Problems with phase constraints . . . . . . . . . .. 360
7.4.4 Optimal switching of observation channels. . . . .. 362
7.5 Observation control problem with constrained number of the
observation instants ................. 366
7.5.1 Solution of the observation control problem 368
7.5.2 Example of the radar system control. . . . 372
8 Appendix. Differential equations with measures 377
8.1 Auxiliary results ........... . 377
8.1.1 Functions of bounded variation ...... . 377
xii Contents
8.1.2 Measures and functions of bounded variation . . .. 381
8.1.3 The Lebesgue-Stieltjes integral . . . . . . . . . . .. 386
8.1.4 Some results concerning the Lebesgue-Stieltjes inte-
gration ................. 390
8.2 Linear differential equations with a measure . . . . . . .. 399
8.2.1 Some general remarks and definitions ....... 399
8.2.2 Scalar linear differential equation with a measure. 400
8.2.3 Multi-dimensional case. . . . . . . . . . . 404
8.2.4 Generalized Gronwall-Bellman inequality 408
8.2.5 Matrix equations . . . . . . . . . . . . . 411
8.3 Nonlinear differential equations with a measure . 412
8.3.1 The solution existence and uniqueness . . 412
8.3.2 Dependence on parameters and initial conditions 416
Bibliography 425