Study on Optimal Controls and Differential Games via the Viscosity Solution Theory

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(1) Showing the equivalence of boundary conditions between Dirichlet and state-constraint types, we characterize the value function of minimum arrival time problems, and give a representation formula of it. We also prove the equivalence between the property of semicontinuous viscosity solution and the dynamic programming principle.(2) We set up a typical differential game "pursuit-evasion" problem to characterize the value function. We also obtain the convergence of semi-discretized approximate value functions.(3) We construct ε-optimal controls for state constraint problems directly from the Hamilton-Jacobi equations without using semi-discrete approximations.(4) We study fully nonlinear second order uniformly elliptic PDEs with superlinear growth for first derivatives, and with possibly discontinuous coefficients and inhomogenious terms. When the growth order is less than quadratic, by the Aleksandrov-Bakelman-Pucci (ABP for short) maximum principle and Caffarelli's argument, we obtain the Harnack inequality to show the Holder continuity. In the quadaratic case, we give a counter-example for the ABP maximum principle while we present a sufficient condition so that the ABP maximum principle holds, and obtain the existence of L^p-viscosity solutions for Dirichlet problems.(5) We characterize viscosity solutions for fully discontinuous limit PDEs of variational problems with various energies having equivalent norms.(6) We obtain locally W^<2,∞> estimates on solutions of obstacle problems arising in mathematical finance to construct optimal policy via one-dimensional Ito formula.