Theory and applications of viscosity solutions

粘度溶液的理论与应用

• 批准号: 09440067
• 负责人: ISHII Hitoshi
• 金额: $ 8.58万
• 依托单位: TOKYO METROPOLITAN UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 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
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09440067/
• 关键词: viscosity solutions; Hamilton-Jacobi equations; degenerate elliptic equations; curvature flow; optimal control; stochastic optimal control; homogenization; semicontinuous viscosity solutions; ガウス曲率流; 確率微分方程式; 不変領域; 摩耗のモデル; Lax-Hopfの公式; 非線形偏微分

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.

所得结果总结如下。 1. 我们介绍了一种通过相应的 Hamilton-Jacobi 方程的粘度解来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 方程半连续粘度解的唯一性和存在性定理是在价值函数半连续时表征最优控制中价值函数的基本工具。我们在希尔伯特空间中建立了类似于巴伦-詹森定理的定理。 3.考虑了遍历控制中的Hamilton-Jacobi方程，并通过相应遍历最优控制的一种值函数来表征Hamilton-Jacobi方程粘度解的存在性。 4. 在半连续粘度解的 Barron-Jensen 理论中，哈密顿量的凸性是一个关键假设。我们引入了具有非凸哈密顿量的哈密顿-雅可比方程的半连续粘度解的概念，该方程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贝尔曼方程解的可解性、唯一性、光滑性及其奇异极限与微分博弈的关系。 6. 提出了凸体高斯曲率流的几何逼近格式并证明了其收敛性。 7. 证明了受控随机微分方程关于紧集的不变性与相应贝尔曼方程的粘性解紧集的限制性质之间的等价性。 8. 研究了凸集高斯曲率流的等待时间现象，证明了如果初始曲面上的一点的两个主曲率消失，则该点的等待时间为正。

项目成果

石井仁司: "Waiting time effects for Gauss curvature flows"Indiana Univ. Math. J.. 48・1. 311-334 (1999)

石井仁：“高斯曲率流的等待时间效应”印第安纳大学数学杂志 48・1（1999）

石井 仁司: "Hopf-Lax formulas for semicontinuous data"Indiana Univ. Math. J.. 48・3. 994-1035 (1999)

石井仁：“半连续数据的 Hopf-Lax 公式”印第安纳大学数学杂志 48・3（1999）

冨田義人: "Unbounded viscosity solutions of nonlinear second order PDE's"Advances in Math. Sciences Appl.. 10・2(未定).

富田义人：“非线性二阶偏微分方程的无界粘度解”数学科学应用进展。10・2（待定）。

儀我美一: "On strong maximum principle and large time behavior of generalized mean curvature flow with the Neumann boundary condition"J. Differential Equations. 154. 107-131 (1999)

Yoshikazu Giga：“关于具有诺依曼边界条件的广义平均曲率流的强极大原理和大时间行为”J. 微分方程。 154. 107-131 (1999)

O. Alvarez, E. N. Barron, H. Ishii: "Hopf-Lax formulas for semicontinuous data"Indiana Univ. Math. J.. 48. 994-1035 (1999)

O. Alvarez、E. N. Barron、H. Ishii：“半连续数据的 Hopf-Lax 公式”印第安纳大学。