  Using a compactness-uniqueness approach, we show that the Fattorini criterion
implies the exact controllability of general compactly perturbed controlled
linear systems. We then apply this perturbation result to obtain new
controllability results for systems governed by partial differential equations.
Notably, we combine it with the fictitious control method to establish the
exact controllability of a cascade system of coupled wave equations by a
reduced number of controls and with the same control time as the one for a
single wave equation. We also combine this perturbation result with
transmutation techniques to prove the null controllability in arbitrarily small
time of a one-dimensional non diagonalizable system of coupled heat equations
with as many controls as equations.
