Mathematical Sciences: Boundary Control Problems for Linear and Non-Linear Partial Differential Equations and Riccati Equations

这个研究项目是围绕在现代技术应用中出现的线性和非线性偏微分方程的最优性、规律性、稳定性或反馈稳定化问题展开的。研究的三个主要领域，由一个共同的主题结合在一起，是：相互作用的结构和相关的智能材料，由耦合偏微分方程系统建模，通常是不同类型的，在边界可能有状态变量的耦合；某些弹性壳的线性和非线性模型的分析和控制理论，如球帽，以及具有其他几何形状的壳；对具有高内阻尼且受一定边界控制作用的弹性模型进行了分析和控制。在研究的核心是二次优化问题，导致Riccati理论描述的最优解的综合；偏微分方程解在内部和边界处的正则性线性和非线性偏微分方程的反馈镇定。本研究的动机是在结构优化、稳定和控制方面所遇到的问题。这些问题的核心是根据预先选定的定量标准对某些物理系统的演化进行优化的概念。这类系统通常可以并且被建模为包含控制变量的线性或非线性偏微分方程系统，这些控制变量将根据所涉及的特定优化准则进行选择。在本项目中，这种方法被应用于复杂相互作用的研究，例如热/弹性或流体/结构耦合。目标是根据偏微分方程建立适当的数学模型，并根据对数学模型的仔细分析设计最优控制策略。＊＊＊

9504822 Lasiecka/Triggiani This research project is centered around issues of optimality, regularity, stability or feedback stabilization, for classes of linear and nonlinear partial differential equations which arise in modern technological applications. The three main areas of the study, bound together by a common theme, are: interacting structures and related smart materials , modeled by systems of coupled partial differential equations, typically of different types, with possible couplings of state variables at the boundary; analysis and control theory of linear and nonlinear models of certain elastic shells, such as a spherical caps, but also shells having other geometries; and analysis and control theory of elastic models which exhibit high internal damping and which are acted on by certain boundary controls. At the core of the study are problems of quadratic optimization, leading to a Riccati theory describing the synthesis of the optimal solution; regularity of solutions of partial differential equations in the interior and at the boundary; and feedback stabilization of linear and nonlinear partial differential equations. This study is motivated by problems arising in structural optimization, stabilization and control. At the core of these issues lies the notion of optimization, in accordance with some pre-selected quantitative criterion, of the evolution of some physical system. Such systems often can, and are, modeled as systems of linear or nonlinear partial differential equations involving control variables which are to be selected according to the particular optimization criteria involved. In this project, such an approach is applied to the study of complex interactions such as arise, for example, in thermal/elastic or fluid/structure couplings. The goal is to develop appropriate mathematical models based on partial differential equations and to design optimal control strategies based on a careful analysis of the mathematical models. ***