Applications to the optimal control and differential game via the viscosity solution theory

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(1) We obtain the uniqueness and representation formula for lower semicontinuous (lsc for short) viscosity solutions of Hamilton-Jacobi (HJ for short) equations with degenerate coefficients.This is a "lsc" version of a result of Ishii and Ramaswamy.(2) We show the uniqueness of viscosity solutions of pursuit-evasion games of the state constraint (SC for short) problem and the convergence of time-discrete approximations to the value function. These results generalize those of Alziary.(3) We show the uniqueness of lsc viscosity solutions for the minimum time problem. To show the equivalence between the Dirichlet type boundary condition and the SC type one, we prove in general that a function is a lsc viscosity solution of the associated PDE if and only if it satisfies the Dynamic Programming Principle. This extends the corresponding result by P.-L. Lions for continuous viscosity solutions.(4) We construct ε-optimal feedback controls for the SC problem. To construct them, we use the information from the associated HJ equation. Furthermore, we need the inf-convolution approximations, the monotone formula for super-differentials and the convergence of value functions for SC problems of smaller domains.(5) We obtain (a) the comparison principle and (b) interior Holder continuity of viscosity solutions of fully nonlinear, second-order, uniformly elliptic PDEs which involve superlinear growth terms for the gradient. The result (a) extends that of Caffarelli, Crandall, Kocan and Swiech while the result (b) generalizes that of Caffarelli.To show (b), we need a generalized Alexandroff-Bakelman-Pucci maximum principle, a precise construction of classical supersolutions of the associated extremal PDEs, and a modified cube decomposition lemma.