Nonnegative control of finite-dimensional linear systems

Lohéac, Jérôme; Trélat, Emmanuel; Zuazua, Enrique

doi:10.1016/j.anihpc.2020.07.004

Lohéac, Jérôme ¹ ; Trélat, Emmanuel ² ; Zuazua, Enrique ^3,^4,⁵

¹ a Université de Lorraine, CNRS, CRAN, F-54000 Nancy, France
² b Sorbonne Université, CNRS, Université de Paris, Inria, Laboratoire Jacques-Louis Lions (LJLL), F-75005 Paris, France
³ c Chair in Applied Analysis, Alexander von Humboldt-Professorship, Department of Mathematics, Friedrich-Alexander-Universiät Erlangen-Nürnberg, 91058 Erlangen, Germany
⁴ d Chair of Computational Mathematics, Fundación Deusto, University of Deusto, 48007 Bilbao, Basque Country, Spain
⁵ e Departamento de Matematicas, Universidad Autonoma de Madrid, 28049 Madrid, Spain

Annales de l'I.H.P. Analyse non linéaire, mars – avril 2021, Tome 38 (2021) no. 2, pp. 301-346.

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Résumé
Abstract

Dans cet article, nous considérons la contrôlabilité d'un système linéaire avec des contrôles positifs. Malgré la condition du rang de Kalman, la condition de positivité des contrôles peut conduire à l'existence d'un temps minimal de contrôlabilité strictement positif. Lorsque tel est le cas, nous démontrons que si la matrice du système de contrôle possède une valeur propre réelle, alors il existe dans l'espace des mesures de Radon positives, un contrôle en le temps minimal et ce contrôle est nécessairement une somme finie de masse de Dirac. De plus, lorsque toutes les valeurs propres de la matrice sont réelles, ce contrôle est unique et le nombre de masses de Dirac le constituant est d'au plus la moitié de la dimension de l'espace d'état. Nous particularisons ces résultats sur l'exemple de l'équation de la chaleur unidimensionnelle, avec des contrôles frontières de type Dirichlet, discrétisée en espace et nous proposons quelques simulations numériques.

We consider the controllability problem for finite-dimensional linear autonomous control systems with nonnegative controls. Despite the Kalman condition, the unilateral nonnegativity control constraint may cause a positive minimal controllability time. When this happens, we prove that, if the matrix of the system has a real eigenvalue, then there is a minimal time control in the space of Radon measures, which consists of a finite sum of Dirac impulses. When all eigenvalues are real, this control is unique and the number of impulses is less than half the dimension of the space. We also focus on the control system corresponding to a finite-difference spatial discretization of the one-dimensional heat equation with Dirichlet boundary controls, and we provide numerical simulations.

MR Zbl

DOI : 10.1016/j.anihpc.2020.07.004

Keywords: Minimal time, Nonnegative control, Dirac impulse
Mots-clés : Temps minimal, Contrôles positifs, Impulsions de Dirac

Affiliations des auteurs :

Lohéac, Jérôme ¹ ; Trélat, Emmanuel ² ; Zuazua, Enrique ^{3, 4, 5}

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     title = {Nonnegative control of finite-dimensional linear systems},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {301--346},
     publisher = {Elsevier},
     volume = {38},
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Lohéac, Jérôme; Trélat, Emmanuel; Zuazua, Enrique. Nonnegative control of finite-dimensional linear systems. Annales de l'I.H.P. Analyse non linéaire, mars – avril 2021, Tome 38 (2021) no. 2, pp. 301-346. doi : 10.1016/j.anihpc.2020.07.004. http://www.numdam.org/articles/10.1016/j.anihpc.2020.07.004/

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