Exact Boundary Controllability of a System of Mixed Order with Essential Spectrum
Résumé
We address in this work the exact boundary controllability of a linear hyperbolic system of the form $u^{\prime\prime}+Au=0$ with $u=\left(u_{1},u_{2}\right)^{T}$ posed in $(0,T)\times(0,1)^{2}$. A denotes a self-adjoint operator of mixed order that usually appears in the modelization of a linear elastic membrane shell. The operator A possesses an essential spectrum which prevents the exact controllability from holding uniformly with respect to the initial data $\left(u^{0},u^{1}\right)$. We show that the exact controllability holds by one Dirichlet control acting on the first variable $u_{1}$ for any initial data $\left(u^{0},u^{1}\right)$ generated by the eigenfunctions corresponding to the discrete part of the spectrum of A. The proof relies on a suitable observability inequality obtained by way of a full spectral analysis and the adaptation of an Ingham-type inequality for the Laplacian in two spatial dimensions. This work provides a nontrivial example of a system controlled by a number of controls strictly lower than the number of components. Some numerical experiments illustrate our study.
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