Generalized Solutions of First Order PDEs: The Dynamical Optimization Perspective (Systems & Control: Foundations & Applications)

Subbotin, Andrei I.

ISBN 10: 0817637400 / ISBN 13: 9780817637408

Verlag: Birkhäuser, 1994

Beschreibung

Bestandsnummer des Verkäufers I-9780817637408

Bibliografische Details

• Titel: Generalized Solutions of First Order PDEs: The Dynamical Optimization Perspective (Systems & Control: Foundations & Applications)
• Verlag: Birkhäuser
• Erscheinungsjahr: 1994
• Sprache: Englisch
• ISBN-10: 0817637400
• ISBN-13: 9780817637408
• Einband: Hardcover
• Zustand: New

Über diesen Titel

Inhaltsangabe

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].

Von der hinteren Coverseite

In this book, the authors present another approach that can be considered as a non-classical method of characteristics, according to which the generalized solution ( called the minimax solution ) is assumed to be weak invariant with respect to the so-called characteristic inclusions.