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Journal Articles Discrete and Continuous Dynamical Systems - Series A Year : 2023

Hautus-Yamamoto criteria for approximate and exact controllability of linear difference delay equations

Abstract

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.
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Dates and versions

hal-03827918 , version 1 (24-10-2022)
hal-03827918 , version 2 (14-05-2023)

Identifiers

Cite

Yacine Chitour, Sébastien Fueyo, Guilherme Mazanti, Mario Sigalotti. Hautus-Yamamoto criteria for approximate and exact controllability of linear difference delay equations. Discrete and Continuous Dynamical Systems - Series A, 2023, 43 (9), pp.3306--3337. ⟨10.3934/dcds.2023049⟩. ⟨hal-03827918v2⟩
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