In this paper, we study the linear wave equation on an n-dimensional spatial domain. We show that there is a boundary triplet associated to the undamped wave equation. This enables us to characterise all boundary conditions for which the undamped wave equation possesses a unique solution non-increasing in the energy. Furthermore, we add boundary inputs and outputs to the system, thus turning it into an impedance conservative boundary control system.

Crouzeix stated the following conjecture in [Integral Equations Operator Theory, 48 (2004), pp. 461-477]: For every n x n matrix A and every polynomial p, |p(A)| le 2 max_z in W(A)|p(z)|, where W(A) is the numerical range of A. We show that the conjecture holds in its strong, completely bounded form, i.e., where p above is allowed to be any matrix-valued polynomial, for all tridiagonal 3 x 3 matrices with constant main diagonal, [smallmatrix a b_1 0, c_1 a b_2, 0 c_2 a], a,b_k,c_k in C, or equivalently, for all complex 3 x 3 matrices with elliptic numerical range and one eigenvalue at the center of the ellipse. We also extend the main result of Choi in [Linear Algebra Appl., 438 (2013), pp. 3247-3257] slightly.

We extend the internal model principle for systems with boundary control and boundary observation, and construct a robust controller for this class of systems. However, as a consequence of the internal model principle, any robust controller for a plant with infinite-dimensional output space necessarily has infinite-dimensional state space. We proceed to formulate the approximate robust output regulation problem and present a finite-dimensional controller structure to solve it. Our main motivating example is a wave equation on a bounded multidimensional spatial domain with force control and velocity observation at the boundary. In order to illustrate the theoretical results, we construct an approximate robust controller for the wave equation on an annular domain and demonstrate its performance with numerical simulations.