  The paper deals with the controllability of finite-dimensional linear
difference delay equations, i.e., dynamics for which the state at a given time
$t$ is obtained as a linear combination of the control evaluated at time $t$
and of the state evaluated at finitely many previous instants of time
$t-\Lambda_1,\dots,t-\Lambda_N$. Based on the realization theory developed by
Y.Yamamoto for general infinite-dimensional dynamical systems, we obtain
necessary and sufficient conditions, expressed in the frequency domain, for the
approximate controllability in finite time in $L^q$ spaces, $q \in [1,
+\infty)$. We also provide a necessary condition for $L^1$ exact
controllability, which can be seen as the closure of the $L^1$ approximate
controllability criterion. Furthermore, we provide an explicit upper bound on
the minimal times of approximate and exact controllability, given by
$d\max\{\Lambda_1,\dots,\Lambda_N\}$, where $d$ is the dimension of the state
space.
