Generalized Solutions of First Order PDEs: The Dynamical Optimization Perspective / Edition 1

by Andrei I. Subbotin

Overview

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].

Product Details

ISBN-13:	9780817637408
Publisher:	Birkhäuser Boston
Publication date:	12/22/1994
Series:	Systems & Control: Foundations & Applications
Edition description:	1995
Pages:	314
Product dimensions:	6.10(w) x 9.25(h) x 0.24(d)

Table of Contents

I Generalized Characteristics of First-Order PDE’s.- II Cauchy Problems for Hamilton-Jacobi Equations.- III Differential Games.- IV Boundary-Value Problems for First-Order PDE’s.- A1 Justification of the Classical Method of Characteristics.- A2 Multifunctions.- A3 Semicontinuous Functions.- A4 Convex Functions.- A5 Contingent Tangent Cones, Directional Derivatives, Subdifferentials.- A6 On a Property of Subdifferentials.- A7 Differential Inclusions.- A8 Criteria for Weak Invariance.

Editorial Reviews

"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

From the Publisher