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In this paper, we study the relative controllability of linear differenceequations with multiple delays in the state by using a suitable formula for thesolutions of such systems in terms of their initial conditions, their controlinputs, and some matrix-valued coefficients obtained recursively from thematrices defining the system. Thanks to such formula, we characterize relativecontrollability in time $T$ in terms of an algebraic property of thematrix-valued coefficients, which reduces to the usual Kalman controllabilitycriterion in the case of a single delay. Relative controllability is studiedfor 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 fordifferent delays in terms of their rational dependence structure, proving thatrelative controllability for some delays implies relative controllability forall delays that are "less rationally dependent" than the original ones, in asense that we make precise. Finally, we provide an upper bound on the minimalcontrollability time for a system depending only on its dimension and on itslargest delay.

Relative controllability of linear difference equations