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

通过粘度解理论在最优控制和微分博弈中的应用

基本信息

批准号：
09640242
负责人：
KOIKE Shigeaki
金额：
$ 2.3万
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：
1997
资助国家：
日本
起止时间：
1997 至 1999
项目状态：
已结题

项目摘要

(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.

(1)获得了退化系数Hamilton-Jacobi（简称HJ）方程下半连续（简称lSC）粘性解的唯一性和表示公式，这是石井和拉马斯瓦米结果的“lSC”版本。(2)我们证明了状态约束问题的追逃对策粘性解的唯一性和时间离散逼近的收敛性。这些结果推广了Alziary的结果。(3)我们证明了最小时间问题的lsc粘性解的唯一性。为了证明Dirichlet型边界条件与SC型边界条件的等价性，我们证明了一个函数是相关偏微分方程的lsc粘性解的充要条件是它满足动态规划原理.这将相应的结果推广到P. - L. Lions提供连续粘度解决方案。(4)我们构造了SC问题的ε-最优反馈控制。为了构造它们，我们使用来自相关HJ方程的信息。此外，我们还需要下卷积近似，超微分的单调公式和较小区域SC问题的值函数的收敛性。(5)我们得到了包含梯度超线性增长项的完全非线性二阶一致椭圆型偏微分方程粘性解的比较原理和内部保持器连续性.结果（a）推广了Caffarelli，Crandall，Kocan和Swiech的结果，而结果（B）推广了Caffarelli的结果，为了证明结果（B），我们需要一个推广的Alexandroff-Bakelman-Pucci极大值原理，一个相关极值偏微分方程经典上解的精确构造，以及一个修正的立方分解引理.