Systems & Control

Foundations & Applications

By: Andrei I. Subbotin

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Hardcover | 22 December 1994

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332 Pages

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Hamilton-Jacobi equations and other types of partial differential equa tions of the first order are dealt with in many branches of mathematics, mechanics, and physics. These equations are usually nonlinear, and func tions vital for the considered problems are not smooth enough to satisfy these equations in the classical sense. An example of such a situation can be provided by the value function of a differential game or an optimal control problem. It is known that at the points of differentiability this function satisfies the corresponding Hamilton-Jacobi-Isaacs-Bellman equation. On the other hand, it is well known that the value function is as a rule not everywhere differentiable and therefore is not a classical global solution. Thus in this case, as in many others where first-order PDE's are used, there arises necessity to introduce a notion of generalized solution and to develop theory and methods for constructing these solutions. In the 50s-70s, problems that involve nonsmooth solutions of first order PDE's were considered by Bakhvalov, Evans, Fleming, Gel'fand, Godunov, Hopf, Kuznetzov, Ladyzhenskaya, Lax, Oleinik, Rozhdestven ski1, Samarskii, Tikhonov, and other mathematicians. Among the inves tigations of this period we should mention the results of S.N. Kruzhkov, which were obtained for Hamilton-Jacobi equation with convex Hamilto nian. A review of the investigations of this period is beyond the limits of the present book. A sufficiently complete bibliography can be found in [58, 126, 128, 141].

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"Subbotin's book is a very valuable addition to the literature."
-- Mathematical Reviews

"The book is printed excellently and clearly. The explanations concerning the content are distinguished of high correctness and equipped with numerous examples and illustrations."
-- ZAA

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Non-Fiction Mathematics Calculus & Mathematical Analysis Differential Calculus & Equations Applied Mathematics Reference, Information & Interdisciplinary Subjects Research & Information Information theory Cybernetics & Systems Theory Optimisation

Introduction
Generalized Characteristics of First-Order PDE's	p. 1
The Classical Method of Characteristics	p. 1
Characteristic Inclusions	p. 8
Upper and Lower Semicontinuous Solutions	p. 15
Criteria of Weak Invariance for Minimax Solutions	p. 24
Piecewise Smooth Solutions	p. 41
Cauchy Problems for Hamilton-Jacobi Equations	p. 55
Minimax Solutions of Hamilton-Jacobi Equations	p. 55
Uniqueness of Minimax Solution of Cauchy Problem for Hamilton-Jacobi Equation	p. 62
Existence of Minimax Solution of Cauchy Problem for Hamilton-Jacobi Equation	p. 69
Uniqueness under Weakened Assumptions	p. 85
Constructive and Numerical Methods	p. 96
Differential Games	p. 115
Basic Notions of the Theory of Differential Games	p. 115
Proof of Existence of Value Function of Differential Game	p. 125
Stable Bridges and Extremal Strategies	p. 134
Some Remarks	p. 147
Mixed Feedback Strategies and Counter-Strategies	p. 162
Constructive and Numerical Methods of the Theory of Differential Games	p. 276
Boundary-Value Problems for First-Order PDE's	p. 201
Cauchy Problems for Hamilton Jacobi Equations with Additional Conditions in the Form of Inequalities	p. 201
Discontinuous Solutions of Dirichlet-Type Boundary-Value Problem	p. 219
Time-Optimal Differential Games	p. 237
Piecewise-Linear Approximations to Minimax Solutions of Hamilton Jacobi Equations	p. 251
App. A1 Justification of the Classical Method of Characteristics	p. 263
App. A2 Multifunctions	p. 266
App. A3 Semicontinuous Functions	p. 268
App. A4 Convex Functions	p. 269
App. A5 Contingent Tangent Cones, Directional Derivatives, Subdifferentials	p. 270
App. A6 On a Property of Subdifferentials	p. 276
App. A7 Differential Inclusions	p. 279
App. A8 Criteria for Weak Invariance	p. 283
Bibliography	p. 291
Index	p. 311
Table of Contents provided by Blackwell. All Rights Reserved.

Systems & Control

Foundations & Applications

At a Glance

Hardcover

Industry Reviews

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