Approximate and exact controllability of linear difference equations

Chitour, Yacine; Mazanti, Guilherme; Sigalotti, Mario

Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 93-142.

Cet article traite de la contrôlabilité approchée et exacte de l’équation aux différences linéaire $x\left(t\right)={\sum }_{j=1}^N{A}_{j}x(t-{\Lambda }_j)+Bu\left(t\right)$ dans $L^2$, avec $x\left(t\right)\in {ℂ}^d$ et $u\left(t\right)\in {ℂ}^m$, en s’appuyant sur une formule de représentation de la solution $x$ en termes de la condition initiale, du contrôle $u$ et de coefficients matriciels appropriés. Lorsque ${\Lambda }_1,\cdots ,{\Lambda }_N$ sont commensurables, les contrôlabilités approchée et exacte sont équivalentes et peuvent être caractérisées par un critère de type Kalman. Cet article s’attache à donner des caractérisations des contrôlabilités approchée et exacte sans hypothèse de commensurabilité. Dans le cas d’un système bi-dimensionnel avec deux retards, nous obtenons une caractérisation explicite des contrôlabilités approchée et exacte en termes des paramètres du problème. Pour le cas général, nous prouvons que la contrôlabilité approchée de zéro vers les états constants est équivalente à la contrôlabilité approchée dans $L^2$. Le résultat correspondant à la contrôlabilité exacte est vrai au moins pour les systèmes bi-dimensionnels avec deux retards.

In this paper, we study approximate and exact controllability of the linear difference equation $x\left(t\right)={\sum }_{j=1}^N{A}_{j}x(t-{\Lambda }_j)+Bu\left(t\right)$ in $L^2$, with $x\left(t\right)\in {ℂ}^d$ and $u\left(t\right)\in {ℂ}^m$, using as a basic tool a representation formula for its solution in terms of the initial condition, the control $u$, and some suitable matrix coefficients. When ${\Lambda }_1,\cdots ,{\Lambda }_N$ are commensurable, approximate and exact controllability are equivalent and can be characterized by a Kalman criterion. This paper focuses on providing characterizations of approximate and exact controllability without the commensurability assumption. In the case of two-dimensional systems with two delays, we obtain an explicit characterization of approximate and exact controllability in terms of the parameters of the problem. In the general setting, we prove that approximate controllability from zero to constant states is equivalent to approximate controllability in $L^2$. The corresponding result for exact controllability is true at least for two-dimensional systems with two delays.

Classification : 39A06, 93B05, 93C65
Keywords: Linear difference equation, delay, approximate controllability, exact controllability
Mot clés : Équation aux différences linéaire, retard, contrôlabilité approchée, contrôlabilité exacte

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