Local stabilization of fluid-structure models

Abstract: We shall address the problem of stabilizing systems coupling the incompressible Navier-Stokes equations with plate or beam equations modeling elastic structures located at the boundary of the domain occupied by the fluid. The control acts only in the beam/plate equation.  We have already studied such systems in 'Feedback stabilization of a fluid-structure model,  SIAM J. Control Optim. 2010'.  Here we consider similar problems but with mixed boundary conditions for the fluid equations and in the case where the steady state around which we look for the stabilization is non zero. The analysis of the underlying linearized system is much more tricky in that case.  For regular initial data, small enough,  we prove the existence of controls stabilizing the coupled system with an arbitrarily prescribed exponential decay rate around non zero steady states. We shall show some numerical simulations illustrating the efficiency of the feedback control.   This is a joint work with M. Ndiaye and M. Fourni´e.