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

9504822 Lasiecka/Triggiani这项研究项目围绕着现代技术应用中出现的线性和非线性偏微分方程类的最优性、正则性、稳定性或反馈镇定问题。这项研究的三个主要领域由一个共同的主题结合在一起：相互作用的结构和相关的智能材料，由通常不同类型的耦合偏微分方程组模拟，在边界上可能存在状态变量的耦合；某些弹性壳的线性和非线性模型的分析和控制理论，例如球帽，但也具有其他几何形状的壳；以及具有高内部阻尼并受某些边界控制作用的弹性模型的分析和控制理论。研究的核心问题是二次优化问题，导致了描述最优解的合成的Riccati理论；偏微分方程解在内部和边界上的正则性；以及线性和非线性偏微分方程解的反馈镇定。本研究是针对结构优化、稳定和控制中出现的问题展开的。这些问题的核心是，根据某种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。这样的系统通常可以，并且正在被建模为线性或非线性偏微分方程组系统，该系统涉及根据所涉及的特定优化标准来选择的控制变量。在这个项目中，这种方法被应用于研究复杂的相互作用，例如，在热/弹性或流体/结构耦合中产生的相互作用。目标是根据偏微分方程建立适当的数学模型，并在仔细分析数学模型的基础上设计最优控制策略。***

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. ***