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

(1)证明了Dirichlet和状态约束型边界条件的等价性，刻画了最小到达时间问题的值函数，并给出了它的一个表示式。并证明了半连续粘性解的性质与动态规划原理的等价性。(2)建立了一个典型的微分对策“追逐-回避”问题来刻画值函数。我们直接从哈密顿-雅可比方程出发构造了状态约束问题的ε-最优控制，而不使用半离散近似。(4)研究了具有超线性增长的一阶导数、可能具有间断系数和非齐次项的完全非线性二阶一致椭圆型偏微分方程解。当增长级小于二次时，利用Aleksandrov-Bakelman-Pucci(简称ABP)极大值原理和Caffarelli论证，得到了证明Holder连续性的Harnack不等式。在二次情形下，我们给出了ABP极大值原理的一个反例，同时给出了ABP极大值原理成立的充分条件，并得到了Dirichlet问题的L p-粘性解的存在性。(5)刻画了具有等价范数的各种能量的变分问题的全间断极限偏微分方程解的粘性解。(6)通过一维Ito公式，我们得到了数学金融中出现的障碍问题的解的局部W^lt；2，∞&gt；估计，从而构造了最优策略。

(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.