×

Optimal control via nonsmooth analysis. (English) Zbl 0874.49002

CRM Proceedings & Lecture Notes. 2. Providence, RI: American Mathematical Society (AMS). ix, 153 p. (1993).
This monograph stemmed from lectures given by the author at the Summer School on Control Theory held in 1992 at the Centre de Recherches Mathématiques de l’Université de Montréal. The objective is to present a unified treatment of deterministic problems in calculus of variations or optimal control by using techniques and results from nonsmooth analysis.
The book can be described by dividing it into four parts. Chapter 1 mainly contains motivations from calculus of variations (existence of solutions, necessary and/or sufficient conditions for optimality, examples) and optimal control (Hamiltonians, maximum principle); the author also motivates the unavoidable work of handling nonsmooth functions and presents the value function which is going to play a key role in the subsequent chapters.
Chapter 2 is aimed at proving existence of solutions to constrained variational problems by the direct method (the problem of Bolza for example). Measure theory and integration (concerning also multifunctions) are central here; some technical results like the measurable selection theorem are explained with detailed proofs.
The next block consists of Chapters 3, 4 and 5. They contain the basics of nonsmooth analysis, in a Hilbert space setting, in view of studying variational problems or just questions from analysis. One can find there variational principles (the classical one by Ekeland but also the more recent one by Borwein and Preiss), analytical and geometrical definitions of the proximal subdifferential and generalized gradient (or subdifferential) in Clarke’s sense, calculus rules going with these concepts, etc. These three chapters can be read independently of the others, and the results therein used in other areas than the ones focused on in the book.
Chapters 6 and 7 are the more advanced ones; they are devoted to dynamic optimization, with analogues of Euler-Lagrange equations and the canonical equations of Hamilton for nonsmooth problems of optimal control, properties and differential characterization of the value function in optimal control, etc. This part somewhat overlaps with F. H. Clarke’s 1989 monograph [“Methods of dynamic and nonsmooth optimization” (1989; Zbl 0696.49003)].
Each chapter closes with a list of exercises and references for further reading; it is a pity that there is no index to identify places where notions are introduced and recurrently used.
The style of exposition is lively, engaging, incisive sometimes; a real and successful effort has been made to render the material accessible to nonspecialists of the subject.

MSC:

49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
49J52 Nonsmooth analysis
93-02 Research exposition (monographs, survey articles) pertaining to systems and control theory
90C30 Nonlinear programming

Citations:

Zbl 0696.49003
PDFBibTeX XMLCite