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.

该提案涉及一些非线性偏微分方程（PDE），它们与最优控制理论、博弈论、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来理解这些非线性偏微分方程解的性质。本研究的主要对象之一是一类非凸哈密尔顿-雅可比方程，它是两人零和微分博弈的基本方程。实现其解的更深层次性质（梯度的奇异结构、大时间平均值等）将有助于设计快速数值方法以准确地逼近解，以及为游戏中的参与者设计最优策略。所提出的项目是（i）继续开发一种新方法来获得 Hamilton-Jacobi 方程和相关问题的解的大时间行为，（ii）发现博弈论解释和动力学性质 一些弱耦合系统的解，(iii) 研究一些 Hamilton-Jacobi 方程的齐次化，以及 (iv) 获得偏微分方程方法来研究摩擦系数消失的 Langevin 方程的渐近极限。这些主题由广泛不同的非线性问题组成，但它们都满足极大值原理，因此允许粘度解。 Crandall-Lions 粘度解理论在过去三十年中得到了广泛的发展，包括解的存在性、唯一性、稳定性以及与微分博弈、前向传播、均质化理论、最优控制和弱 KAM 理论的一些联系。然而，粘度解的许多有趣的性质，例如规律性、动力学性质、梯度冲击结构和解的博弈论解释，仍然远未得到很好的理解。 PI建议开发一些新方法来研究(i)-(iv)，这有望为粘度解决方案领域带来新的视角和见解。 (i)-(iv)所使用的数学工具由非线性伴随法（对偶法）、动力系统、水平集法、最优控制理论和博弈论等技术组成。