Singular asymptotic expansion of the exact control for a linear model of the Rayleigh beam

The Petrowsky type equation $y_{tt}^{\text{eps}}+\text{eps}{y}_{xxxx}^{\text{eps}}-y_{xx}^{\text{eps}}=0$, $\text{eps}>0$ encountered in linear beams theory is null controllable through Neumann boundary controls. Due to the boundary layer of size of order $\sqrt{\text{eps}}$ occurring at the extremities, these boundary controls get singular as $\text{eps}$ goes to $0$. Using the matched asymptotic method, we describe the boundary layer of the solution $y^{\text{eps}}$ then derive a rigorous second order asymptotic expansion of the control of minimal $L^2-$norm, with respect to the parameter $\text{eps}$. 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.