Generalized solutions of first-order PDEs : the dynamical optimization perspective
Author(s)
Bibliographic Information
Generalized solutions of first-order PDEs : the dynamical optimization perspective
（Systems & control）
Birkhäuser, c1995
Available at 27 libraries
Note
Includes bibliographical references and index
Description and Table of Contents
- Volume
-
ISBN 9780817637408
Description
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].
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.
- Volume
-
ISBN 9783764337407
Description
Hamilton-Jacobi equations and other types of partial differential equations of the first order are dealt with in many branches of mathematics, mechanics and physics. As a rule, functions vital for the considered problems are not smooth enough to satisfy these equations in the classical sense. Thus, there arises the need to introduce a notion of a generalized solution and to develop theory and methods for constructing these solutions. This text presents an approach to partial differential equations that can be considered as a non-classical method of characteristics, according to which the generalized solution (the minimax solution) is assumed to be flow invariant with respect to the so-called characteristic inclusions. The research on minimax solutions employs methods of the theory of differential games, dynamical optimization and nonsmooth analysis. At the same time, this research has contributed to the development of these new branches of mathematics. The book is intended as a self-contained exposition of the theory of minimax solutions. It includes existence and uniqueness results, examples of modelling and applications to the theory of control and differential games.
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