Infinite Dimensional Linear Control Systems (eBook)

The Time Optimal and Norm Optimal Problems

eBook Download: EPUB

2005 | 1. Auflage
332 Seiten
Elsevier Science (Verlag)
978-0-08-045734-5 (ISBN)

Infos

For more than forty years, the equation y'(t) = Ay(t) + u(t) in Banach spaces has been used as model for optimal control processes described by partial differential equations, in particular heat and diffusion processes. Many of the outstanding open problems, however, have remained open until recently, and some have never been solved. This book is a survey of all results know to the author, with emphasis on very recent results (1999 to date).

The book is restricted to linear equations and two particular problems (the time optimal problem, the norm optimal problem) which results in a more focused and concrete treatment. As experience shows, results on linear equations are the basis for the treatment of their semilinear counterparts, and techniques for the time and norm optimal problems can often be generalized to more general cost functionals.

The main object of this book is to be a state-of-the-art monograph on the theory of the time and norm optimal controls for y'(t) = Ay(t) + u(t) that ends at the very latest frontier of research, with open problems and indications for future research.

Key features:

? Applications to optimal diffusion processes.
? Applications to optimal heat propagation processes.
? Modelling of optimal processes governed by partial
differential equations.
? Complete bibliography.
? Includes the latest research on the subject.
? Does not assume anything from the reader except
basic functional analysis.
? Accessible to researchers and advanced graduate
students alike

? Applications to optimal diffusion processes.
? Applications to optimal heat propagation processes.
? Modelling of optimal processes governed by partial
differential equations.
? Complete bibliography.
? Includes the latest research on the subject.
? Does not assume anything from the reader except
basic functional analysis.
? Accessible to researchers and advanced graduate
students alike

Inhalt

Cover 1
Contents 11
Preface 7
Introduction 13
Finite dimensional systems: the maximum principle 13
Finite dimensional systems: existence and uniqueness 21
Infinite dimensional systems 28
Systems with strongly measurable controls, I 39
The reachable space and the bang-bang property 39
Reversible systems 48
The reachable space and its dual, I 59
The reachable space and its dual, II 68
The maximum principle 77
Vanishing of the costate and nonuniqueness in norm optimality 89
Vanishing of the costate for time optimal controls 99
Singular norm optimal controls 108
Singular norm optimal controls and singular functionals 120
Systems with strongly measurable controls, II 129
Existence and uniqueness of optimal controls 129
The weak maximum principle and the time optimal problem 137
Modeling of parabolic equations 146
Weakly singular extremals 155
More on the weak maximum principle 164
Convergence of minimizing sequences to optimal controls 175
Optimal control of heat propagation 185
Modeling of parabolic equations 185
Adjoints 192
Adjoint semigroups 199
The reachable space 203
The reachable space and its dual, I 209
The reachable space and its dual, II 219
The maximum principle 227
Existence, uniqueness and stability of optimal controls 237
Examples and applications 243
Optimal control of heat diffusion 255
Modeling of parabolic equations 255
The reachable space and its dual, I 264
The reachable space and its dual, II 270
The maximum principle 278
Existence of optimal controls uniqueness and stability of supports
Examples and applications 297
Appendix 307
Self adjoint operators, I 307
Self adjoint operators, II 313
Related research 317
References 321
Notation and subject index 331

Textauszug

Preface

One of the first infinite dimensional control systems to come under scrutiny was

′t=Ayt+ut,y0=ζ

  (1)

where y(t) takes values in a Banach space E and A is the infinitesimal generator of a strongly continuous semigroup S(t). Research on the time optimal problem started in the early sixties and branched into other optimal control problems. However, many basic questions were unsolved at the end of the last century. Since then, some new results have clarified the panorama but the subject is still in need of proofs of, or counterexamples to many natural conjectures and it remains a live research area with many actual and potential applications.

In spite of being, in a sense, the simplest infinite dimensional control system, the equation (1) models some important control processes such as those described by the parabolic equation

ytx∂t=Aytx+utx

  (2)

where A is an elliptic operator in the space variables x = (x1,x2,…,xm) in a domain Ω of m-dimensional Euclidean space m; the domain of A is restricted by boundary conditions. The control u(t, x) satisfies bounds of the type

Ω|utx|pdx≤C

  (3)

for some p, 1 ≤ p < ∞, or

utx|≤C.

  (4)

These bounds determine the state space E in which (2) is modeled. For the bound (3) the space is E = Lp(Ω). For the uniform bound (4) we take E = L∞(Ω), (or, rather =CΩ¯. The most physically significant cases are (4) for heat processes and (3) with p = 1 for diffusions. On the other hand, p = 2 leads to the simplest mathematics since the state space L2(Ω) is a Hilbert space.

We consider two optimal control problems for the equation (1). Both of them include a target condition

T=y¯.

  (5)

In the norm optimal problem we minimize

.sup0≤t≤T‖ut‖

among all solutions of (1) satisfying the initial condition y(0) = ζ and the target condition (5); the control interval 0 ≤ t ≤ T is fixed. In the time optimal problem the controls satisfy a fixed bound such as

.sup0≤t≤T‖ut‖≤1,

and we minimize T subject to the initial and target condition.

The time optimal problem received privileged attention from the very start of control theory, but this has been less the case for the norm optimal problem. It was known for a long time that time optimality implies norm optimality, but that the two problems are far from equivalent in the infinite dimensional setting seems to have been realized much more recently. However, there are many situations (determined by conditions on the semigroup S(t) or on the target ¯) where time and norm optimality are essentially equivalent.

Most of this book deals with the relation among time and norm optimality and Pontryagin’s maximum principle

T−t*z,u¯t〉=max‖u‖≤1〈ST−t*z,u

  (6)

(||u|| ≤ minimum norm for the norm optimal problem). The maximum principle with z E* = dual of E is a necessary and (almost) sufficient condition for time and norm optimality in finite dimension.1 The finite dimensional theory extends to the equation (1) when S(t)E = E for t > 0 (in particular when S(t) is a group) but the similarities with the finite dimensional case end here. In general, special assumptions on the target ¯ are needed to make (6) a necessary condition for optimality (with z in a space larger than E*) and, conversely, special assumptions on z are needed to make (6) a sufficient condition for optimality of ¯t. In fact, singular optimal controls (those that do not satisfy Pontryagin’s maximum principle, or satisfy it only in a weak form) are the main actors in various places of this monograph.

Much of the material is independent of the maximum principle. Under suitable conditions on z, (6) implies the bang-bang principle

ut‖=1a.e.

  (7)

but (7) can be also be proved without intercession of the maximum principle for time optimal controls. Other results (some depending on the maximum principle, some not) include various well posedness properties of control problems, that is, continuous dependence of optimal controls on parameters of the system such as the initial and target conditions.

This monograph is organized as follows. The first two sections of Chapter 1 contain a survey of some finite dimensional results with an outline of infinite dimensional systems in the third. The aim is reveal that some infinite dimensional results are descendants of finite dimensional theorems. In some cases, however the “family resemblance” is slight, and many other results have no counterpart in finite dimension. We have included some references to the early history of infinite dimensional control theory in 1.3.

Chapter 2 and Chapter 3 deal with the system (1) in an arbitrary Banach space E with a view towards the modeling of partial differential equations such as (2) in Lp(Ω) for 1 < p < ∞. However, we do also other equations; for instance, some of the most interesting examples in 2.6 and 2.7 use the “proto-hyperbolic” equation yt(t, x) = —yx(t, x). These results suggest that a systematic study of the maximum principle (or, rather, of its interpretation) for equations of hyperbolic type would be worth undertaking, but this is not attempted here.

Chapter 4 is on the modeling of the equation (2) in Ω¯. Due to existence requirements, the control space must be expanded to L∞(Ω) and we can request only weak measurability of the controls with respect to t; this corresponds to driving (2) with controls in L∞((0,T) × Ω).

Chapter 5 is on the modeling of the equation (2) in L1(Ω). The control space L1((0,T) × Ω) places us in a adverse existence situation, thus we must replace it by a space of (weakly measurable) controls taking values in the space Ω¯ of Borcl measures in ¯.

There is an obvious parallelism between the two models in Chapters 4 and Chapter 5, so much so that one can translate results from one case to the other using a “replacement chart” where Ω¯ is replaced by L1(Ω), L∞(Ω) is replaced by Ω¯… and so on. However, the similarity does not go all the way. Although both the geometries of L∞(Ω) and Σ(Ω) are devoid of smoothness, the control space L∞(Ω) allows uniqueness results for optimal controls not unlike those in smooth spaces, while in the space Ω¯ uniqueness breaks down completely both in the time optimal and the norm optimal problems. In a sort of compensation, in the Ω¯ setting the bang-bang principle (7) is, in certain situations, essentially sufficient for both time optimality and norm optimality, a result that is not known to hold in any other space.

The treatments in Chapters 4 and 5 could have been unified by means of the theory of Phillips adjoints, but the gain in brevity and conciseness would not outweigh the additional insight that each of the parallel theories afford.

The first two sections of Chapter 6 deal with some results that are known to hold for a very restricted class, self adjoint semigroups in Hilbert spaces, and examine the possibility of generalizations to other semigroups and Banach spaces. In the last section we include some recent references on the time and norm optimal problems as well as on problems not treated in this book but related to the material in one way or another, among...

Produkt-Details

• Erscheint lt. Verlag: 12.7.2005
• Mitarbeit: Herausgeber (Serie): H.O. Fattorini
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Technik
• ISBN-10: 0-08-045734-7 / 0080457347
• ISBN-13: 978-0-08-045734-5 / 9780080457345