  In this paper, we study the relative controllability of linear difference
equations with multiple delays in the state by using a suitable formula for the
solutions of such systems in terms of their initial conditions, their control
inputs, and some matrix-valued coefficients obtained recursively from the
matrices defining the system. Thanks to such formula, we characterize relative
controllability in time $T$ in terms of an algebraic property of the
matrix-valued coefficients, which reduces to the usual Kalman controllability
criterion in the case of a single delay. Relative controllability is studied
for solutions in the set of all functions and in the function spaces $L^p$ and
$\mathcal C^k$. We also compare the relative controllability of the system for
different delays in terms of their rational dependence structure, proving that
relative controllability for some delays implies relative controllability for
all delays that are "less rationally dependent" than the original ones, in a
sense that we make precise. Finally, we provide an upper bound on the minimal
controllability time for a system depending only on its dimension and on its
largest delay.
