Viability theory

Jean-Pierre Aubin (auth.)9780817649098, 0817649093, 0817635718, 9780817635718, 3764335718, 9783764335717, 9780817649104

This work examines viability theory and its applications to control theory and differential games. The emphasis is on the construction of feedbacks and dynamical systems by myopic optimization methods. Systems of first-order partial differential inclusions, whose solutions are feedbacks, are constructed and investigated. Basic results are then extended to the case of fuzzy control problems, distributed control problems, and control systems with delays and memory.

Aimed at graduate students and research mathematicians, both pure and applied, this book offers specialists in control and nonlinear systems tools to take into account general state constraints. Viability theory also allows researchers in other disciplines—artificial intelligence, economics, game theory, theoretical biology, population genetics, cognitive sciences—to go beyond deterministic models by studying them in a dynamical or evolutionary perspective in an uncertain environment.

“The book is a compendium of the state of knowledge about viability…Mathematically, the book should be accessible to anyone who has had basic graduate courses in modern analysis and functional analysis…The concepts are defined and many proofs of the requisite results are reproduced here, making the present book essentially self-contained.” (Bulletin of the AMS)

“Because of the wide scope, the book is an ideal reference for people encountering problems related to viability theory in their research…It gives a very thorough mathematical presentation. Very useful for anybody confronted with viability constraints.” (Mededelingen van het Wiskundig Genootschap)

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Table of contents :
Front Matter….Pages i-xxv
Introduction….Pages 1-9
Outline of the Book….Pages 11-17
Viability Theorems for Ordinary and Stochastic Differential Equations….Pages 19-52
Set-Valued Maps….Pages 53-75
Viability Theorems for Differential Inclusions….Pages 77-118
Viability Kernels and Exit Tubes….Pages 119-155
Invariance Theorems for Differential Inclusions….Pages 157-198
Regulation of Control Systems….Pages 199-234
Smooth and Heavy Viable Solutions….Pages 235-273
Partial Differential Inclusions of Tracking Problems….Pages 275-314
Lyapunov Functions….Pages 315-350
Miscellaneous Viability Issues….Pages 351-375
Viability Tubes….Pages 377-399
Functional Viability….Pages 401-423
Viability Theorems for Partial Differential Inclusions….Pages 425-450
Differential Games….Pages 451-484
Back Matter….Pages 485-545