DIFFERENTIAL RICCATI EQUATION FOR THE ACTIVE CONTROL OF A PROBLEM IN STRUCTURAL ACOUSTICS

In this paper, we provide results concerning the optimal feedback cont rol of a system of partial differential equations which arises within the context of modeling a particular fluid/structure interaction seen in structural acoustics, this application being the primary motivation for our work. This system consists of two coupled PDEs exhibiting hyp erbolic and parabolic characteristics, respectively, with the control action being modeled by a highly unbounded operator. We rigorously jus tify an optimal control theory for this class of problems and further characterize the optimal control through a suitable Riccati equation. This is achieved in part by exploiting recent techniques in the area o f optimization of analytic systems with unbounded inputs, along with a local microanalysis of the hyperbolic part of the dynamics, an analys is which considers the propagation of singularities and optimal trace behavior of the solutions.