Riccati Equations Arising in Acoustic Structure Interactions with Curved Walls

Structural acoustic control problems are considered. The main aim is to reduce a pressure/noise in an acoustic cavity by an appropriate activation of piezoceramic devices. The physical model is comprised of an acoustic chamber with flexible (elastic) walls to which piezoceramic devices are attached. The devices play the role of actuators and sensors. The mathematical description of the model is governed by a coupled system of partial differential equations (PDE\u27s) involving the wave equation coupled with the dynamic shell equation (modeling the wall). The goal of this paper is to present new results on optimal control problems with smart controls. The control algorithm is constructed in a feedback form via a solution of a suitable Riccati type equation. The main technical/mathematical difficulty of the problem is related to the fact that the control operators are unbounded. This leads, in general, to the unbounded gain operators and optimal synthesis of the control function which is defined in a very weak sense only. However, for the problem at hand, we show that in spite of the unboundedness of the control operators, the feedback gains are bounded and the optimal synthesis is fully meaningful. This is due to the regularizing effects of shell dynamics which are partially propagated into the hyperbolic component of the structure