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Generalized Solutions of First Order PDEs - The Dynamical Optimization Perspective (Paperback, Softcover reprint of the original 1st ed. 1995)
Loot Price: R2,884
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Generalized Solutions of First Order PDEs - The Dynamical Optimization Perspective (Paperback, Softcover reprint of the original 1st ed. 1995)
Series: Systems & Control: Foundations & Applications
Expected to ship within 10 - 15 working days
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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].
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