1999 Fiscal Year Final Research Report Summary
Theory and applications of viscosity solutions
| Project/Area Number |
09440067
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (B)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
解析学
|
|---|
| Research Institution | TOKYO METROPOLITAN UNIVERSITY |
|---|
Principal Investigator |
ISHII Hitoshi Tokyo Metropolitan University, Science, Professor, 理学研究科, 教授 (70102887)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
TOMITA Yoshihito Kobe University of Mercantile Marine, Mercantile Marine Science, Professor, 商船学部, 教授 (50031456)
GIGA Yoshikazu Hokkaido University, Science, Professor, 大学院・理学研究科, 教授 (70144110)
MOCHIZUKI Kiyoshi Tokyo Metropolitan University, Science, Professor, 大学院・理学研究科, 教授 (80026773)
ISHII Katsuyuki Kobe University of Mercantile Marine, Mercantile Marine Science, Associate Professor, 商船学部, 助教授 (40232227)
KOIKE Shigeaki Saitama University, Science, Associate Professor, 理学部, 助教授 (90205295)
|
|---|
| Project Period (FY) |
1997 – 1999
|
| Keywords | viscosity solutions / Hamilton-Jacobi equations / degenerate elliptic equations / curvature flow / optimal control / stochastic optimal control / homogenization / semicontinuous viscosity solutions |
|---|
| Research Abstract |
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.
|
