Analysis and Control Theory for Moving Boundary and Nonlinear Phenomena in Interactive Partial Differential Equations

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The theme of this research project is to conduct mathematical and computational studies on classes of partial differential equations (PDE) with moving boundaries. The first part of the project mathematically describes the phenomenon of acoustic sound waves within a chamber whose flexible walls are allowed to move freely as time evolves. For such acoustic wave dynamics, the goal is to determine which pre-assigned chamber geometries will give rise to an interior acoustic field that is as "quiet" as possible. That is to say, because of the analytically pre-determined chamber configuration, the acoustic pressure will be minimized and external noise optimally reduced. The second part of the project focuses on structure-fluid interactions as they occur in nature and engineering, for example the nucleus of a cell interacting with the cellular cytoplasm. In this example, the wall of the nucleus might be described by structural PDE, and the cytoplasm mathematically modeled by fluid flow PDE. The goal of the project is to develop and analyze more realistic models that comprise both fluid and structure PDE dynamics. The first part of the project will address the feasibility of stabilizing interior acoustic wave dynamics on a time-evolving bounded domain through the means of prescribing, in advance, the evolution of the geometry. That is, as time gets large, the prescribed moving boundary feedback law should cause the acoustic wave solutions to either decay or approach a certain set of finite energy states. Since the associated wave energy is majorized by wave boundary trace quantities that resemble velocity feedback mechanisms, this "geometry-control" problem might be viewed as a moving boundary analog to Neumann boundary stabilization of the wave equation on a fixed domain. The second part of the project will consider linear and nonlinear structure-fluid PDE models in which the structural components are mathematically modeled by physically relevant but quite complex Reissner-Mindlin-Timshenko (RMT) plate systems. By decoupling these structure-fluid dynamics, the associated pressure variables are eliminated in a nonstandard manner, intrinsic to the coupling mechanisms in place. Having resolved the well-posedness, the study will consider an associated optimal control problem that will reveal certain ocular dynamics. In addition, the question of global exact controllability for nonlinear von Karman plate dynamics will be investigated. The objective is to establish the exact steering property with respect to finite energy states of arbitrary size, not only target states within a small neighborhood of the origin. This control process will be simulated numerically and the dependence of the controllability time on the size of the initial and target states will be examined.