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

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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. ***