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

Abstract

The Petrowsky type equation y ε tt + εy ε xxxx − y ε xx = 0, ε > 0 encountered in linear beams theory is null controllable through Neumann boundary controls. Due to the boundary layer of size of order √ ε 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 y ε then derive a rigorous second order asymptotic expansion of the control of minimal L 2 −norm, with respect to the parameter ε. In particular, we recover that the leading term of the expansion is a null Dirichlet control for the limit hyperbolic wave equation, in agreement with earlier results due to J-.L. Lions in the eighties. Numerical experiments support the analysis.