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

The Petrowsky type equation y e tt + ey e xxxx − y e xx = 0, e > 0 encountered in linear beams theory is null controllable through Neumann boundary controls. Due to the boundary layer of size of order √ e occurring at the extremities, these boundary controls get singular as e goes to 0. Using the matched asymptotic method, we describe the boundary layer of the solution y e then derive a rigorous second order asymptotic expansion of the control of minimal L 2 −norm, with respect to the parameter e. 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 du to J-.L. Lions in the eighties. Numerical experiments support the analysis.