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

9504822 Lasiecka/Triggiani本研究项目围绕最优性、正则性、稳定性或反馈稳定性问题，针对现代技术应用中出现的线性和非线性偏微分方程。 该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，在边界上可能有状态变量的耦合;某些弹性壳体（如球形盖）以及具有其他几何形状的壳体的线性和非线性模型的分析和控制理论;以及具有高内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。 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合成;正则性的解决方案的偏微分方程在内部和边界;和反馈稳定的线性和非线性偏微分方程。 本研究的动机是在结构优化，稳定和控制中出现的问题。 这些问题的核心在于优化的概念，根据一些预先选择的定量标准，一些物理系统的演变。 这样的系统通常可以并且被建模为包括控制变量的线性或非线性偏微分方程的系统，所述控制变量将根据所涉及的特定优化标准来选择。 在这个项目中，这种方法被应用到复杂的相互作用的研究，如出现，例如，在热/弹性或流体/结构耦合。 其目标是开发适当的数学模型的基础上偏微分方程和设计最佳控制策略的基础上仔细分析的数学模型。 ***

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. ***