Mathematical Sciences: Control of Variable Coefficient Partial Differential Equations: Elastic Bodies and Schrodinger Equations

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9501051 Horn This project focuses on boundary controllability and stabilization of elastic systems and has the two primary objectives. The first is to address the question of controllability of a Kirchhoff plate model with nonconstant coefficients in the principal part of the differential operator and with control acting as a moment on the boundary. Because of the close relationship between the Kirchhoff plate and a time-dependent Schroedinger equation with potential, as a first step, a general technique which relies on local smoothing properties of the solution is used to establish exact boundary controllability the Schroedinger equation with continuous coefficients. These results will then be utilized to establish exact boundary controllability of the Kirchhoff plate equation. The second major goal is to resolve the issue of controllability for the equations of three- dimensional linear elasticity, an issue which naturally arises when moving from two- dimensional plate models to mathematical models of general three-dimensional elastic bodies. Results currently available assume very strict geometric conditions on the domain even when the control is acting on the entire boundary. To examine the boundary controllability problem for the fully coupled system while eliminating these stringent geometric restrictions, an approach based on energy estimates and microlocal analysis techniques will be used. Derivation of trace regularity estimates for the solution of the system of linear elasticity will be a critical step in the process. Extensions of boundary controllability results from linear, constant coefficient distributed parameter systems to systems with nonconstant coefficients are mf great importance both mathematically and physically. Accurate mathematical modeling of many important physical systems requires that spatial inhomogeneities be admitted into the model and that the physical parameters in the model be permitted to vary with time. For this reason, it is important to develop a control theory based on these more accurate models of the physical world. In particular, a control theory for such models based on boundary sensors and actuators is sought, since one of the most important applications of such a theory is to the problem of controlling large flexible structures, where the natural locations for both measurements and controls are the joints and outer edges of the structure. Because both elastic plate models and equations of linear elasticity may be used to model components, such as airfoils and solar panels, of many structures, understanding how such elements can be controlled is a significant step to the development of a comprehensive control strategy for complex systems. ***