Generalized Solutions of First Order PDEs: The Dynamical Optimization PerspectiveSpringer Science & Business Media, 29.06.2013 - 314 Seiten 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]. |
Inhalt
Discontinuous Solutions of DirichletType | 18 |
6 | 25 |
Piecewise Smooth Solutions | 41 |
Cauchy Problems for HamiltonJacobi Equations | 54 |
55 | 69 |
Uniqueness under Weakened Assumptions | 85 |
Differential Games | 115 |
Differential Games | 177 |
TimeOptimal Differential Games | 237 |
PiecewiseLinear Approximations to Minimax Solutions | 252 |
Appendix | 263 |
A4 Convex Functions | 269 |
A6 On a Property of Subdifferentials | 276 |
A8 Criteria for Weak Invariance | 283 |
291 | |
311 | |
Andere Ausgaben - Alle anzeigen
Generalized Solutions of First Order PDEs: The Dynamical Optimization ... Andrei I. Subbotin Keine Leseprobe verfügbar - 2013 |
Generalized Solutions of First Order PDEs: The Dynamical Optimization ... Andrei I. Subbotin Keine Leseprobe verfügbar - 2013 |
Generalized Solutions of First Order PDEs: The Dynamical Optimization ... Andrei I. Subbotin Keine Leseprobe verfügbar - 1994 |
Häufige Begriffe und Wortgruppen
According Arg max assume Cauchy problem characteristic inclusions condition x(to construction continuous function controlled system convex D+u(x defined definition denote diam differential game differential inclusion Du(x epigraph epiu equality equivalent estimate formula function satisfies Hamilton-Jacobi equations Hamiltonian hypograph implies inequality initial condition invariant with respect Isaacs-Bellman equation Lipschitz condition locally bounded lower semicontinuous lower solution measurable function method of characteristics minimax solution mixed feedback strategies multifunction Note obtain optimal payoff functional player Q positive number problem 7.1 proof of Theorem Proposition Recall respect to differential result S(to satisfies condition satisfies the conditions semicontinuous function Sol(to solution of equation solution of problem stable bridge ti+1 tion u-stable u(to u(xo upper lower upper semicontinuous upper solution v(to valid value function vector viscosity solutions weakly invariant x(ti y(to