Exact Boundary Controllability of a System of Mixed Order with Essential Spectrum

Abstract

Abstract. We address in this work the exact boundary controllability of a linear hyperbolic system of the form u ′′ + Au =0withu =( u 1 ,u 2 )T posed in (0,T) × (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 u 0 ,u 1 . We show that the exact controllability holds by one Dirichlet control acting on the first variable u 1 for any initial data u 0 ,u 1 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 Inghamtype 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.