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

Yacine Chitour; Sébastien Fueyo; Guilherme Mazanti; Mario Sigalotti

doi:10.3934/dcds.2023049

Article Contents

2023, Volume 43, Issue 9: 3306-3337. Doi: 10.3934/dcds.2023049

This issue Previous Article Non-injectivity of infinite interval exchange transformations and generalized Thue-Morse sequences Next Article Sobolev regularity theory for the non-local elliptic and parabolic equations on $ C^{1,1} $ open sets

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

1.
Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des signaux et systèmes, 91190 Gif-sur-Yvette, France
2.
Sorbonne Université, Inria, CNRS, Laboratoire Jacques-Louis Lions (LJLL), 75005 Paris, France
3.
Université Paris-Saclay, CNRS, CentraleSupélec, Inria, Laboratoire des signaux et systèmes, 91190 Gif-sur-Yvette, France

^*Corresponding author: Sébastien Fueyo
^*Corresponding author: Sébastien Fueyo

Received: November 2022

Revised: April 2023

Early access: May 18, 2023

Published online: May 18, 2023

The third author is supported by ANR PIA funding: ANR-20-IDEES-0002.

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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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Mathematics Subject Classification: Primary: 39A06, 93B05; Secondary: 93C05.

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