Mathematical Sciences: Infinite Dimensional Nonlinear Programming Problems

该奖项支持偏微分方程式和泛函微分方程式领域的数学研究。本文主要研究无限维非线性规划理论在由Navier-Stokes方程、反应扩散方程和非线性波动方程等方程控制的系统的最优控制问题中的应用。其目的是利用Hamilton-Jacoby方程得到松弛控制领域的存在性结果、庞特里亚金最优控制类型的必要条件、次优控制序列的数值收敛结果和反馈解。对于包括状态约束的问题，将在分布式和边界控制的情况下进行工作。包含奇异圆弧的问题的正则化和哈密顿量在自由到达时间问题中的作用也将被研究。偏微分方程式是建立物理世界数学模型的基础。数学分析的作用与其说是创建方程，不如说是提供有关解的定性和定量信息。这可能包括回答有关唯一性、平稳性和成长性的问题。此外，分析经常开发出近似解的方法和对这些近似的精度的估计。

This award supports mathematical research in the areas of partial differential equations and functional differential equations. The work focuses on the applications of infinite dimensional nonlinear programming theory to optimal control problems for systems governed by equations such as the Navier-Stokes equations, reaction-diffusion equations and nonlinear wave equations. The object is to obtain existence results in the realm of relaxed controls, necessary conditions for optimal controls of the type of Pontryagin's maximum principle, convergence results for numerically obtained sequences of suboptimal controls and feedback solutions by means of Hamilton-Jacoby equations. Work will be done both in the case of distributed and boundary control for problems that include sate constraints. Regularization of problems containing singular arcs and the role of the Hamiltonian in free arrival time problems will also be examined. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations.