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تسجيل مستخدم جديدSingular asymptotic expansion of the exact control for a linear model of the Rayleigh beam
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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 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.
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