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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 throughNeumann boundary controls. Due to the boundary layer of size of order$\sqrt{\eps}$ occurring at the extremities, these boundary controls getsingular as $\eps$ goes to $0$. Using the matched asymptotic method, wedescribe the boundary layer of the solution $y^\eps$ then derive a rigoroussecond order asymptotic expansion of the control of minimal $L^2-$norm, withrespect to the parameter $\eps$. In particular, we recover that the leadingterm of the expansion is a null Dirichlet control for the limit hyperbolic waveequation, in agreement with earlier results due to J-.L. Lions in the eighties.Numerical experiments support the analysis.

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