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

Carlos Castro (1) , Arnaud Munch (2, 3)

1 UPM - Universidad Politécnica de Madrid
2 UCA [2017-2020] - Université Clermont Auvergne [2017-2020]
3 LMBP - Laboratoire de Mathématiques Blaise Pascal

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

Mots clés: Singular controllability; Boundary layers; Asymptotic analysis; Numerical experiments

Domaines: Optimisation et contrôle [math.OC]

Carlos Castro, Arnaud Munch. Singular asymptotic expansion of the exact control for a linear model of the Rayleigh beam. 2019. ⟨hal-02169764v1⟩