{"product_id":"generalized-solutions-of-first-order-pdes-the-dynamical-optimization-perspective-von-andrei-i-subbotin","title":"Generalized Solutions of First Order PDEs","description":"\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\u003cdiv class=\"aw-variant-hidden-subtitle-div\" id=\"aw-variant-subtitle-9780817637408\"\u003e\u003ch3\u003eThe Dynamical Optimization Perspective\u003c\/h3\u003e\u003c\/div\u003e\u003cdiv class=\"aw-variant-hidden-subtitle-div\" id=\"aw-variant-subtitle-9781461269205\"\u003e\u003ch3\u003eThe Dynamical Optimization Perspective\u003c\/h3\u003e\u003c\/div\u003e","brand":"Libri","offers":[{"title":"Hardcover - 9780817637408","offer_id":51469265606,"sku":"9780817637408","price":106.99,"currency_code":"EUR","in_stock":true},{"title":"Softcover - 9781461269205","offer_id":39415699439709,"sku":"9781461269205","price":106.99,"currency_code":"EUR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0940\/0622\/files\/41bb548e-6de3-413a-82ec-c33daa565e67.jpg?v=1772170633","url":"https:\/\/shop.autorenwelt.de\/en\/products\/generalized-solutions-of-first-order-pdes-the-dynamical-optimization-perspective-von-andrei-i-subbotin","provider":"Autorenwelt Shop","version":"1.0","type":"link"}