  We analyse the problem of controllability for parameter-dependent linear
finite-dimensional systems. The goal is to identify the most distinguished
realisations of those parameters so to better describe or approximate the whole
range of controls. We adapt recent results on greedy and weak greedy algorithms
for parameter depending PDEs or, more generally, abstract equations in Banach
spaces. Our results lead to optimal approximation procedures that, in
particular, perform better than simply sampling the parameter-space to compute
the controls for each of the parameter values. We apply these results for the
approximate control of finite-difference approximations of the heat and the
wave equation. The numerical experiments confirm the efficiency of the methods
and show that the number of weak-greedy samplings that are required is
particularly low when dealing with heat-like equations, because of the
intrinsic dissipativity that the model introduces for high frequencies.
