Singular asymptotic expansion of the exact control for the perturbed wave equation

Abstract

The Petrowsky type equation ${\mathit{y}}_{\mathit{t}\mathit{t}}^{\mathit{\epsilon }}+\mathit{\epsilon }{\mathit{y}}_{\mathit{x}\mathit{x}\mathit{x}\mathit{x}}^{\mathit{\epsilon }}-{\mathit{y}}_{\mathit{x}\mathit{x}}^{\mathit{\epsilon }}=0$ , $\mathit{\epsilon }\mathrm{>}0$ encountered in linear beams theory is null controllable through Neumann boundary controls. Due to the boundary layer of size of order $\sqrt{\mathit{\epsilon }}$ occurring at the extremities, these boundary controls get singular as ε goes to 0. Using the matched asymptotic method, we describe the boundary layer of the solution ${\mathit{y}}^{\mathit{\epsilon }}$ and derive a rigorous second order asymptotic expansion of the control of minimal weighted ${\mathit{L}}^2$ -norm, with respect to the parameter ε. The weight in the norm is chosen to guarantee the smoothness of the control. In particular, we recover and enrich earlier results due to J.-L. Lions in the eighties showing that the leading term of the expansion is a null Dirichlet control for the limit hyperbolic wave equation. The asymptotic analysis also provides a robust discrete approximation of the control for any ε small enough. Numerical experiments support our study.