Stabilization of a nonlinear structural acoustic interaction

We consider a structural-acoustic wall problem in dimension 3, in which the structural wall is modeled by a 2D Kirchhoff-Boussinesq plate and acoustic medium subject to a boundary damping. The structure subjected to: a nonlinear external forces [Boussinesq] which may lead to blow up of solutions, and the restoring internal forces mitigating the blow up. It will be shown that a suitable calibration between the forces and boundary damping, subject to certain geometric configurations, leads to a wellposed nonlinear control system. For such system, feedback stabilization of solutions [both strong and uniform] is established. The main difficulties and novelty encountered in the problem are the following: (i) the nonlinear effects are not locally Lipschitz, (ii) the system is not dissipative and (iii) the presence of un-observed Neumann data on the interface. To handle these difficulties new methods in dynamical systems and geometric analysis are developed and applied.