  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 $y^\eps$ then derive a rigorous
second order asymptotic expansion of the control of minimal $L^2-$norm, with
respect to the parameter $\eps$. 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.
