Control and stabilization in nonlinear structural acoustic problems

We consider a model arising in structural acoustic problems which is comprised of an acoustic chamber with flexible walls to which piezo-ceramic devices are attached. The deices play the role of actuators and sensors for the model. The mathematical description of the model is governed by a coupled system of partial differential equations involving the wave equation coupled with a nonlinear dynamic shell equation. The main aim is to reduce a pressure/noise in the cabin by the appropriate activation of piezo-ceramic devices. The assumed periodic nature of the disturbance, leads naturally to the formulation of a periodic control problem. This, in turn, is strongly linked to the problem of stabilizability of the original model. Thus, the goal of this paper is to present new results on (i) uniform stabilizability of the structural acoustic model with passive damping applied to the boundary of the acoustic chamber, and (ii) an optimal control problem with \u27smart\u27 controls activated by piezo-ceramic patches creating suitable bending moments in the structure. The control algorithm is constructed in a feedback from via a solution of a suitable Riccati type equation. One of our results shows the boundedness of the feedback gains in spite of the strong unboundedness of the control operators. This is due to the \u27regularizing\u27 effects of shell dynamics which are partially propagated into the \u27hyperbolic\u27 component of the structure