Local Stabilization of a Fluid--Structure System around a Stationary State with a Structure Given by a Finite Number of Parameters

Abstract

We study the stabilization of solutions to a 2d fluid-structure system by a feedback control law acting on the acceleration of the structure. The structure is described by a finite number of parameters. The modelling of this system and the existence of strong solutions has been previously studied in [11]. We consider an unstable stationary solution to the problem. We assume a unique continuation property for the eigenvectors of the adjoint system. Under this assumption, the nonlinear feedback control that we propose stabilizes the whole fluid-structure system around the stationary solution at any chosen exponential decay rate for small enough initial perturbations. Our method reposes on the analysis of the linearized system and the feedback operator is given by a Riccati equation of small dimension. MSC numbers. 35Q30, 74F10, 76D55, 93D15