# American Institute of Mathematical Sciences

March  2013, 2(1): 119-151. doi: 10.3934/eect.2013.2.119

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

 1 Laboratoire de Mathématiques, Université Blaise Pascal (Clermont-Ferrand), UMR CNRS 6620, Campus des Cézeaux, 63177 Aubière

Received  September 2012 Revised  December 2012 Published  January 2013

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 exact controllability does not hold uniformly with respect to the initial data. Thus the numerical computation of controls by the way of dual approach and gradient method 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 this work, we adapt a variational approach introduced in [Pablo Pedregal, Inverse Problems (26) 015004 (2010)] for the wave equation and obtain a robust method of approximation. This new 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. We also consider a relaxed controllability case for which uniform property holds.
Citation: Arnaud Münch. A variational approach to approximate controls for system with essential spectrum: Application to membranal arch. Evolution Equations & Control Theory, 2013, 2 (1) : 119-151. doi: 10.3934/eect.2013.2.119
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