Optimal location of controllers for the one-dimensional wave equation

In this paper, we consider the homogeneous one-dimensional wave equation defined on $(0,\pi)$. For every subset $\omega\subset [0,\pi]$ of positive measure, every $T \geq 2\pi$, and all initial data, there exists a unique control of minimal norm in $L^2(0,T;L^2(\omega))$ steering the system exactly to zero. In this article we consider two optimal design problems. Let $L\in(0,1)$. The first problem is to determine the optimal shape and position of $\omega$ in order to minimize the norm of the control for given initial data, over all possible measurable subsets $\omega$ of $[0,\pi]$ of Lebesgue measure $L\pi$. The second problem is to minimize the norm of the control operator, over all such subsets. Considering a relaxed version of these optimal design problems, we show and characterize the emergence of different phenomena for the first problem depending on the choice of the initial data: existence of optimal sets having a finite or an infinite number of connected components, or nonexistence of an optimal set (relaxation phenomenon). The second problem does not admit any optimal solution except for $L=1/2$. Moreover, we provide an interpretation of these problems in terms of a classical optimal control problem for an infinite number of controlled ordinary differential equations. This new interpretation permits in turn to study modal approximations of the two problems and leads to new numerical algorithms. Their efficiency will be exhibited by several experiments and simulations.