A variational approach to approximate controls for system with essential spectrum : Application to membranal arch

Abstract

We address the numerical approximation of boundary controls for systems of the form $\boldsymbol{y^{\prime\prime}}+\boldsymbol{A_M}\boldsymbol{y}=\boldsymbol{0}$ which models dynamical elastic shell structure. The membranal operator $\boldsymbol{A_M}$ is self-adjoint and of mixed order, so that it possesses a non empty and bounded essential spectrum $\sigma_{ess}(\boldsymbol{A_M})$. Consequently, the controllability does not hold uniformly with respect to the initial data. Thus the numerical computation of controls by the way of dual approachs and gradient methods may fail, even if the initial data belongs to the orthogonal of the space spanned by the eigenfunctions associated with $\sigma_{ess}(\boldsymbol{A_M})$. In that work, we adapt a variational approach introduced in [Pablo Pedregal, \textit{Inverse Problems} (26) 015004 (2010)] for the wave equation and obtain a robust method of approximation. This approach does not require any information on the spectrum of the operator $\boldsymbol{A_M}$. We also show that it allows to extract, from any initial data $(\boldsymbol{y^0},\boldsymbol{y^1})$, a controllable component for the mixed order system. We illustrate these properties with some numerical experiments in the full controllability context as well as a partial controllability one