Hautus-Yamamoto criteria for approximate and exact controllability of linear difference delay equations
Résumé
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−Λ1,…,t−ΛN. 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 Lq spaces, q∈[1,+∞). We also provide a necessary condition for L1 exact controllability, which can be seen as the closure of the L1 approximate controllability criterion. Furthermore, we provide an explicit upper bound on the minimal times of approximate and exact controllability, given by dmax{Λ1,…,ΛN}, where d is the dimension of the state space.
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