{"product_id":"generalized-solutions-of-first-order-pdes-the-dynamical-optimization-perspective-9781461269205","title":"Generalized Solutions of First Order Pdes: The Dynamical Optimization Perspective","description":"\u003cp\u003e • Author(s): Andrei I. Subbotin\u003cbr\u003e • Publisher: Springer\u003cbr\u003e • Publisher Imprint: Birkhauser\u003cbr\u003e • BISAC: Differential Equations - Partial\u003c\/p\u003e\u003cp\u003e\u003c\/p\u003e\u003cp\u003eHamilton-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].\u003c\/p\u003e","brand":"Springer","offers":[{"title":"Paperback","offer_id":45282806431895,"sku":"9781461269205","price":7277.0,"currency_code":"INR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0666\/3471\/1191\/files\/9781461269205.webp?v=1769303238","url":"https:\/\/atlanticbooks.com\/products\/generalized-solutions-of-first-order-pdes-the-dynamical-optimization-perspective-9781461269205","provider":"Atlantic Books","version":"1.0","type":"link"}