  In this paper, we study approximate and exact controllability of the linear
difference equation $x(t) = \sum\_{j=1}^N A\_j x(t - \Lambda\_j) + B u(t)$ in
$L^2$, with $x(t) \in \mathbb C^d$ and $u(t) \in \mathbb C^m$, using as a basic
tool a representation formula for its solution in terms of the initial
condition, the control $u$, and some suitable matrix coefficients. When
$\Lambda\_1, \dotsc, \Lambda\_N$ are commensurable, approximate and exact
controllability are equivalent and can be characterized by a Kalman criterion.
This paper focuses on providing characterizations of approximate and exact
controllability without the commensurability assumption. In the case of
two-dimensional systems with two delays, we obtain an explicit characterization
of approximate and exact controllability in terms of the parameters of the
problem. In the general setting, we prove that approximate controllability from
zero to constant states is equivalent to approximate controllability in $L^2$.
The corresponding result for exact controllability is true at least for
two-dimensional systems with two delays.

Optimization and Control