Flutter stabilization for an unstable hyperbolic flow-plate interaction

Abstract

The aim of these lectures is to present a mathematical theory of control problems arising in flow-structure interactions. From the modeling point of view, these are coupled problems of two dynamicsat an interface. The equations of interest are an unstable linearization of the compressible Eulerequation with a nonlinear plate dynamics. The latter accounts for the effects of possibly largedisplacements, which are typically modeled by the scalar or vectorial von Karman equations oflarge plate deflections. The coupled model is a hybrid interaction between the flow—defined on a3D flow domain—and the plate or shell—defined on a 2D manifold. The unperturbed flow moves in3D with normalized velocity U > 0and excites the structure through the distributed pressure, actingthrough traces on the lower dimensional manifold. As a consequence, the structural displacementsthen perturb the flow. The communication (feedback) between these two systems lies in the heartof the mathematical problem. The described interaction is ubiquitous in nature, with a multitudeof physical applications. Some central examples arise in Fluid Mechanics, Aerospace Applications,Structural Engineering, and Biological Applications.