Computational methods for the fast boundary stabilization of flexible plates

Abstract

The numerical approximation of an efficient active control strategy [6, 7, 3] for undamped flexible plates of arbirary shape controlled from the boundary is considered. A non-standard computational framework is proved to be relevant both for simulation and control synthesis. It enables one to achieve an arbitrarily large decay rate of some weak norm of the system. The proposed formulation proves necessary in so far as a standard numerical approach is shown to fail in a simple case. The paper focusses on general strategies to compute the boundary control that ensure convergence and uniform stability of the closed-loop system. A Galerkin approximation of both the dynamics of the structure and the controller equation is introduced and further numerical integration schemes for modal stresses along the region of controlled are proposed. To this end, branch modes or Poincar- Steklov modes are shown to be good candidates in view of a high-level implementation in a general purpose finite element softwares. The method extends the work explained in [1, 2, 4, 5].