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

基于粘性解理论的最优控制与微分博弈研究

基本信息

批准号：
12640103
负责人：
KOIKE Shigeaki
金额：
$ 2.24万
依托单位：
Saitama University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2002
项目状态：
已结题

项目摘要

(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.

(1)给出了Dirichlet与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，并给出了其表示公式，并证明了半连续粘性解的性质与状态约束型的等价性。动态规划原理。(2)建立了一个典型的微分博弈“追逃”问题来刻画价值函数，并得到了半离散近似价值函数的收敛性。(3)构造了该问题。状态约束问题的 ε 最优控制直接来自 Hamilton-Jacobi 方程，无需使用半离散近似。(4) 我们研究具有一阶导数超线性增长的完全非线性二阶均匀椭圆偏微分方程，并且可能具有不连续系数和非齐次项。当增长阶次小于二次时，由Aleksandrov-Bakelman-Pucci（简称ABP）极大值原理和Caffarelli的论证中，我们得到Harnack不等式来证明Holder连续性。在二次情况下，我们给出了ABP极大值原理的反例，同时给出了ABP极大值原理成立的充分条件，并得到了L^的存在性。狄利克雷问题的 p-粘度解。(5) 我们描述具有等效范数的各种能量的变分问题的完全不连续极限偏微分方程的粘度解。(6) 我们获得局部 W^<2,∞> 估计研究通过一维伊藤公式解决数学金融中出现的障碍问题来构建政策。