Theory and applications of viscosity solutions

所获得的结果总结如下。1.介绍了利用相应的Hamilton-Jacobi方程的粘性解构造状态约束控制问题的近似反馈控制的方法。2.Barron-Jensen关于Hamilton-Jacobi方程半连续粘性解的唯一性和存在性定理是刻画最优控制中值函数半连续的一个基本工具。我们在Hilbert空间中建立了一个类似于Barron-Jensen定理的定理。3.考虑了遍历控制下的Hamilton-Jacobi方程，通过相应的遍历最优控制的一种值函数，给出了该方程粘性解的存在性的一个刻画。4.在半连续粘性解的Barron-Jensen理论中，哈密顿量的凸性是一个关键假设。对于非凸哈密顿量的Hamilton-Jacobi方程，我们引入了半连续粘性解的概念，该粘性解具有良好的唯一性和存在性。5.研究了风险敏感随机控制中Bellman方程解的可解性、唯一性、光滑性及其奇异极限与微分对策的关系。6.给出了凸体高斯曲率流的一种几何逼近格式，并证明了该格式的收敛性。7.证明了受控随机微分方程关于紧集的不变性与相应的Bellman方程粘性解的紧集的约束性质的等价性。8.研究了凸集的高斯曲率流的等待时间现象，证明了如果两个主曲率在初始曲面上的一点上为零，则该点的等待时间为正。

The results obtained are summarized as follows. 1. We introduced a method of constructing an approximate feedback control for state-constraint control problems via viscosity solutions of the corresponding Hamilton-Jacobi equations. 2. The uniqueness and existence theorem due to Barron-Jensen on semicontinuous viscosity solutions of Hamilton-Jacobi equations is a fundamental tool in characterizing value functions in optimal control when the value functions are semicontinuous. We established a theorem similar to the Barron-Jensen theorem in Hilbert spaces. 3. We considered the Hamilton-Jacobi equation in ergodic control and gave a characterization of the existence of viscosity solutions of the Hamilton-Jacobi equation through a kind of value function of the corresponding ergodic optimal control. 4. In the Barron-Jensen theory of semicontinuous viscosity solutions the convexity of Hamiltonians is a key assumption. We introduced a notion of semicontinuous viscosity solution for Hamilton-Jacobi equations with non-convex Hamiltonian for which nice uniqueness and existence properties hold. 5. We studied the solvability, uniqueness, smoothness of solutions of Bellman equations in risk-sensitive stochastic control as well as the relation between its singular limit and a differential game. 6. We introduced a geometric approximation scheme for Gauss curvature flow of a convex body and proved its convergence. 7. We proved the equivalence between the invariance of a controlled stochastic differential equation with respect to a compact set and the restriction property to the compact set of viscosity solutions of the corresponding Bellman equation. 8. We studied the waiting time phenomena for Gauss curvature flow of a convex set and proved that if two principal curvatures vanish at a point on the initial surface then the waiting time of the point is positive.