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Viscosity solutions of Hamilton-Jacobi equations. (Solutions de viscosité des équations de Hamilton-Jacobi.) (French) Zbl 0819.35002
Mathématiques & Applications (Berlin). 17. Paris: Springer-Verlag. ix, 194 p. (1994).
The goal of this book is to present the theory of viscosity solutions for first order Hamilton-Jacobi equations and the applications to deterministic optimal control and similar perturbations. The first half of the book deals with continuous viscosity solutions, while the second half deals with discontinuous solutions. Existence (via Perron’s method), uniqueness, and stability results are presented in a clear manner. Infinite and finite horizon, stopping time, and impulse control problems are described. Problems with state constraints and reflected trajectories are also treated.

MSC:
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35F20 Nonlinear first-order PDEs
35B25 Singular perturbations in context of PDEs
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