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Generalized solutions of firstorder PDEs : the dynamical optimization perspective
Subbotin, A. I. (Andreĭ Izmaĭlovich)Boston : Birkhäuser, c1995.HamiltonJacobi 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 nonclassical method of characteristics, according to which the generalized solution (the minimax solution) is assumed to be flow invariant with respect to the socalled 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 selfcontained 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.

Generalized solutions of first order PDEs : the dynamical optimization perspective
Subbotin, A. I. (Andreĭ Izmaĭlovich)Boston, MA : Springer, 1995.HamiltonJacobi 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 HamiltonJacobiIsaacsBellman 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 firstorder 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 50s70s, 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 HamiltonJacobi 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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Gametheoretical control problems
Krasovskiĭ, N. N. (Nikolaĭ Nikolaevich)New York : SpringerVerlag, c1988.This book is devoted to an investigation of control problems which can be described by ordinary differential equations and be expressed in terms of game theoretical notions. In these terms, a strategy is a control based on the feedback principle which will assure a definite equality for the controlled process which is subject to uncertain factors such as a move or a controlling action of the opponent. Game Theoretical Control Problems contains definitions and formalizations of differential games, existence for equilibrium and extensive discussions of optimal strategies. Formal definitions and statements are accompanied by suitable motivations and discussions of computational algorithms. The book is addessed to mathematicians, engineers, economists and other users of control theoretical and game theoretical notions.
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