Controllability of the Linear One-dimensional Wave Equation with Inner Moving Forces

This paper deals with the numerical computation of distributed null controls for the 1D wave equation. We consider supports of the controls that may vary with respect to the time variable. The goal is to compute approximations of such controls that drive the solution from a prescribed initial state to zero at a large enough controllability time. Assuming a geometric optic condition on the support of the controls, we first prove a generalized observability inequality for the homogeneous wave equation. We then introduce and prove the well-posedness of a mixed formulation that characterizes the controls of minimal square-integrable norm. Such mixed formulation, introduced in [\textit{Cindea and Münch, A mixed formulation for the direct approximation of the control of minimal ${L}^2$-norm for linear type wave equations}], and solved in the framework of the (space-time) finite element method, is particularly well-adapted to address the case of time dependent support. Several numerical experiments are discussed