Singular asymptotic expansion of the exact control for the perturbed wave equation
Résumé
The Petrowsky type equation $y_{tt}^\eps+\eps y_{xxxx}^\eps - y_{xx}^\eps=0$, $\eps>0$ encountered in linear beams theory is null controllable through Neumann boundary controls. Due to the boundary layer of size of order $\sqrt{\eps}$ occurring at the extremities, these boundary controls get singular as $\eps$ goes to $0$. Using the matched asymptotic method, we describe the boundary layer of the solution $y^\eps$ and derive a rigorous second order asymptotic expansion of the control of minimal weighted $L^2-$norm, with respect to the parameter $\eps$. 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 $\eps$ small enough. Numerical experiments support our study.
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