This lecture aims at presenting in a unified way some recent results concerning numerical approximations of controls  for the wave equation.

We will first recall basic facts of control theory in the finite-dimensional setting, and in particular the duality between observability and controllability issues.

Next, we will address these questions for the wave equation in a smooth bounded domain, and explain several ways to show observability for wave equations: one based on a spectral decomposition of the solution in $1$d, one based on micro-local analysis and the propagation of the singularities yielding the celebrated Geometric Control Condition by C. Bardos, G. Lebeau /& J. Rauch, and finally a multiplier approach.

We shall then present the control problem for discrete wave equations. As remarked by R. Glowinski /& al, there are some strongly divergent phenomena when trying to compute numerical approximations of the control. We will explain these pathologies in terms of the spectrum of the discrete wave operator and in terms of discrete Gaussian beams.

We will then turn back to the analysis of the Hilbert Uniqueness Method in the continuous setting. In particular, we will show that, if the initial datum to be controlled is smooth, then the Hilbert Uniqueness Method (or rather a suitably modified version of it) yields a control that provides a controlled trajectory with the same regularity as the initial datum. As a consequence, we will see how to design from these results an algorithm to compute convergent approximations of the controls of the underlying continuous model. We will also exhibit the interest of having observability results for the discretized equations which hold uniformly with the discretization parameters.

Following, it is natural to look for ways to guarantee uniform observability results for the discretized wave equations.

One natural technique to do so is to consider the usual discretized equations and try to reestablish uniform observability results for these discretized equations by reinforcing the observability inequality. This is the most developed path and can be done through filtering conditions, penalizations of the high-frequency components of the solutions, bi-grid approaches, etc. I will also present a new direction that we have developed recently with A. Marica and E. Zuazua which rather consists in adapting the discretization of the equation to the controllability problem at hand.

We will also explain how, in some situations, the effect of the time discretization can be completely decoupled from the effect of the space semi-discretization.