We address in this work the exact boundary controllability of a linear hyperbolic system of the form u'' + Au = 0 with u = (u(1), u(2))(T) posed in (0, T) x (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.