Some new approaches for the study of properties of viscosity solutions

研究粘度溶液性质的一些新方法

• 批准号: 1615944
• 负责人: Hung Tran
• 金额: $ 7.13万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-08-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1615944&HistoricalAwards=false
• 关键词: Some; new; approaches; study; properties

This proposal concerns some nonlinear partial differential equations (PDE), which have deep connections with optimal control theory, game theory, mathematical finance, homogenization theory, and statistical physics. The main goal is to discover new underlying principles and generic methods to understand the properties of solutions of these nonlinear PDEs. One of the main objects of this proposed research is a class of non convex Hamilton--Jacobi equations, which are the fundamental equations for two-person, zero-sum differential games. Achieving deeper properties of their solutions (singular structures of the gradients, large time average, and so forth) will help a lot in the design of fast numerical methods to approximate the solutions accurately and in the design of optimal strategies for the players in the games.The proposed projects are to (i) continue developing a new approach to obtain large time behavior of solutions of Hamilton-Jacobi equations and related problems, (ii) discover game theory interpretation and dynamical properties of solutions of some weakly coupled systems, (iii) study homogenization of some Hamilton-Jacobi equations, and (iv) obtain a PDE approach to study asymptotic limit for the Langevin equation with vanishing friction coefficient. The topics consist of widely different nonlinear problems but they all satisfy maximum principle and hence admit viscosity solutions. The Crandall-Lions theory of viscosity solutions has been developed extensively in the last thirty years including the existence, uniqueness, stability of the solutions as well as some connections to differential games, front propagations, homogenization theory, optimal control, and weak KAM theory. However, many interesting properties of viscosity solutions, such as regularity, dynamical properties, gradient shock structure, and game theory interpretation of solutions, are still far from being well understood. The PI proposes to develop some new approaches to study (i)-(iv), which are expected to bring new perspective and insights to the field of viscosity solutions. The mathematical tools to be used for (i)-(iv) are composed by techniques from the nonlinear adjoint method (duality method), dynamical system, level set method, optimal control theory, and game theory.

该建议涉及一些非线性偏微分方程（PDE），它们与最佳控制理论，游戏理论，数学金融，均质化理论和统计物理学有着深厚的联系。主要目标是发现新的基本原理和通用方法，以了解这些非线性PDE的解决方案的特性。这项拟议研究的主要对象之一是一类非凸汉密尔顿 - 雅各比方程，这是两人零和零差异游戏的基本方程。 Achieving deeper properties of their solutions (singular structures of the gradients, large time average, and so forth) will help a lot in the design of fast numerical methods to approximate the solutions accurately and in the design of optimal strategies for the players in the games.The proposed projects are to (i) continue developing a new approach to obtain large time behavior of solutions of Hamilton-Jacobi equations and related problems, (ii) discover game theory interpretation and dynamical properties of某些弱耦合系统的溶液（iii）研究某些汉密尔顿 - 雅各比方程的均质化，（iv）获得了一种PDE方法，用于研究Langevin方程的渐近限，并以消失的摩擦系数研究。主题由广泛不同的非线性问题组成，但它们都满足最大原理，因此承认粘度解决方案。粘度解决方案的crandall-lions理论在过去的三十年中已经广泛发展，包括存在，独特性，解决方案的稳定性以及与差异游戏的某些连接，前面传播，均质理论，最佳控制和弱KAM理论。但是，粘度解决方案的许多有趣的属性，例如规律性，动力学性能，梯度冲击结构和对解决方案的游戏理论解释，仍然远非被众所周知。 PI建议开发一些新的研究方法（i） - （iv），这些方法有望为粘度解决方案的领域带来新的观点和见解。用于（i） - （iv）的数学工具是由非线性伴随方法（二元方法），动力学系统，级别设置方法，最佳控制理论和游戏理论的技术组成的。