  We address the problem of catching all speed $1$ geodesics of a Riemannian
manifold with a moving ball: given a compact Riemannian manifold $(M,g)$ and
small parameters $\varepsilon>0$ and $v>0$, is it possible to find $T>0$ and an
absolutely continuous map $x:[0,T]\rightarrow M, t\mapsto x(t)$ satisfying
$\|\dot{x}\|_{\infty}\leq v$ and such that any geodesic of $(M,g)$ traveled at
speed $1$ meets the open ball $B_g(x(t),\varepsilon)\subset M$ within time $T$?
Our main motivation comes from the control of the wave equation: our results
show that the controllability of the wave equation can sometimes be improved by
allowing the domain of control to move adequately, even very slowly. We first
prove that, in any Riemannian manifold $(M,g)$ satisfying a geodesic recurrence
condition (GRC), our problem has a positive answer for any $\varepsilon>0$ and
$v>0$, and we give examples of Riemannian manifolds $(M,g)$ for which (GRC) is
satisfied. Then, we build an explicit example of a domain
$X\subset\mathbb{R}^2$ (with flat metric) containing convex obstacles, not
satisfying (GRC), for which our problem has a negative answer if $\epsilon$ and
$v$ are small enough, i.e., no sufficiently small ball moving sufficiently
slowly can catch all geodesics of $X$.
