Hautus--Yamamoto criteria for approximate and exact controllability of linear difference delay equations
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866918091819057152 |
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| author | Chitour, Yacine Fueyo, Sébastien Mazanti, Guilherme Sigalotti, Mario |
| author_facet | Chitour, Yacine Fueyo, Sébastien Mazanti, Guilherme Sigalotti, Mario |
| contents | 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 $t-Λ_1,\dots,t-Λ_N$. 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 $d\max\{Λ_1,\dots,Λ_N\}$, where $d$ is the dimension of the state space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_13590 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Hautus--Yamamoto criteria for approximate and exact controllability of linear difference delay equations Chitour, Yacine Fueyo, Sébastien Mazanti, Guilherme Sigalotti, Mario Optimization and Control 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 $t-Λ_1,\dots,t-Λ_N$. 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 $d\max\{Λ_1,\dots,Λ_N\}$, where $d$ is the dimension of the state space. |
| title | Hautus--Yamamoto criteria for approximate and exact controllability of linear difference delay equations |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2210.13590 |