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Book Chapter10.1007/978-1-4612-0349-0_5

A Theory of Generalized Solutions to First-Order PDEs with an Emphasis on Differential Games

A. I. Subbotin-1993-01-01-Birkhäuser Boston eBooks
2

TL;DRAbstract

It is well known that in many applications connected with boundary value problems and Cauchy problems for Hamilton-Jacobi equations and other types of partial differential equations (PDE) of the first order there exist no classical nonlocal solutions. On the other hand, there exist nonsmooth functions that are crucial for the considered problems (e.g., optimal value functions in control problems, alsoknown as Bellman functions). These functions satisfy the considered equations at each point of differentiability. Thus, there arises a necessity to introduce a notion of generalized solution and to develop the theory and methods for constructing these solutions. Investigations of various approaches to the definition of generalized solutions are dealt with in many papers (see, e.g., the articles [13,28] and the book [32], which contain reviews and bibliography of investigations of 1950s–70s among which we should name the results of S.N.Kruzhkov for the Hamilton-Jacobi equation whose Hamilto

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It is well known that in many applications connected with boundary value problems and Cauchy problems for Hamilton-Jacobi equations and other types of partial differential equations (PDE) of the first order there exist no classical nonlocal solutions. On the other hand, there exist nonsmooth functions that are crucial for the considered problems (e.g., optimal value functions in control problems, alsoknown as Bellman functions). These functions satisfy the considered equations at each point of differentiability. Thus, there arises a necessity to introduce a notion of generalized solution and to develop the theory and methods for constructing these solutions. Investigations of various approaches to the definition of generalized solutions are dealt with in many papers (see, e.g., the articles [13,28] and the book [32], which contain reviews and bibliography of investigations of 1950s–70s among which we should name the results of S.N.Kruzhkov for the Hamilton-Jacobi equation whose Hamilto

Keywords

MathematicsDifferentiable functionBoundary value problemHamilton–Jacobi equationApplied mathematicsInitial value problemHamiltonian (control theory)Hamiltonian system

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