Riccati equations arising in acoustic structure interactions with curved walls
At the begining of the paper, the authors give some motivation for their research coming from various engineering applications which concern the problem of reducing the level of unwanted pressure or noise in a three dimensional acoustic chamber with flexible walls to which some piezoceramic devices are attached. By means of these devices treated as sensors and actuators it is possible to implement suitable control strategies constructing a control algorithm in a feedback form. The mathematical model consists of a coupled system of PDE's involving the wave equation on a three dimensional domain describing the chamber and a dynamic shell equation modeling the wall. Then a cost (penalty) functional is associated in order to minimize the energy of vibrations, as well as, the energy used by controls. Some theorems concerning the stabilization of the uncontrolled system (giving the uniform decay estimates for the energy of the system), as well as the existence of the unique optimal control in a feedback form are provided. The feedback control is constructed via a solution of a suitable Riccati equation. Due to the regularizing effects of shell dynamics the feedback gains are bounded (in spite of unbounded control operators) so the constructed optimal synthesis is fully meaningful.
scientific article; zbMATH DE number 1203649