no. G12
Article de recherche - Systèmes dynamiques, Théorie du contrôle
Controllability and feedback stabilizability in a nonuniform framework
[Contrôlabilité et stabilisabilité par rétroaction dans un cadre non uniforme]
Huerta, Ignacio1  ; Monzón, Pablo2 ; Robledo, Gonzalo3
1Departamento de Matemática, Universidad Técnica Federico Santa María, Casilla 110-V, Valparaíso, Chile
2Facultad de Ingeniería – Universidad de la República, Julio Herrera y Reissig 565, Montevideo, Uruguay
3Departamento de Matemáticas – Universidad de Chile, Casilla 653, Las Palmeras 3425, Ñuñoa – Santiago, Chile
Comptes Rendus. Mathématique, Tome 362 (2024) no. G12, pp. 1667-1692

We propose a new controllability property for linear nonautonomous control systems in finite dimension: the nonuniform complete controllability, which is halfway between the classical Kalman’s properties of complete controllability and uniform complete controllability. This new concept has a strong linkage, as we prove, with the property of nonuniform bounded growth for the corresponding plant. In addition, we also prove that if a control system is nonuniformly completely controllable and its plant (uncontrolled part) has the property of nonuniform bounded growth, then there exist a linear feedback control leading to a nonuniformly exponentially stable closed–loop system.

Nous proposons une nouvelle propriété de contrôlabilité pour des systèmes de contrôle linéaires non autonomes en dimension finie : la contrôlabilité complète non uniforme, qui est à mi-chemin entre les propriétés classiques de contrôlabilité complète et la contrôlabilité complète uniforme établies par Kalman. Ce nouveau concept à un lien étroit, comme nous le prouvons, avec la propriété de croissance bornée non uniforme de la plante correspondante. En outre, il est aussi prouvé que si un système de contrôle est non uniformément complètement contrôlable et sa plante (partie non contrôlée) à la propriété de croissance bornée non uniforme, alors on a un résultat de stabilisation asymptotique par bouclage linéaire ou la convergence est non uniformément exponentielle.

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DOI : 10.5802/crmath.672
Classification : 34D05, 34D09, 93B05, 93D15
Keywords: Linear nonautonomous systems, nonuniform controllability, nonuniform exponential stabilizability, nonuniform bounded growth
Mots-clés : Systèmes linéaires non autonomes, Contrôlabilité non uniforme, Stabilité exponentielle non uniforme, Croissance bornée non uniforme

Huerta, Ignacio  1   ; Monzón, Pablo  2   ; Robledo, Gonzalo  3

1 Departamento de Matemática, Universidad Técnica Federico Santa María, Casilla 110-V, Valparaíso, Chile
2 Facultad de Ingeniería – Universidad de la República, Julio Herrera y Reissig 565, Montevideo, Uruguay
3 Departamento de Matemáticas – Universidad de Chile, Casilla 653, Las Palmeras 3425, Ñuñoa – Santiago, Chile
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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     author = {Huerta, Ignacio and Monz\'on, Pablo and Robledo, Gonzalo},
     title = {Controllability and feedback stabilizability in a nonuniform framework},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1667--1692},
     year = {2024},
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Huerta, Ignacio; Monzón, Pablo; Robledo, Gonzalo. Controllability and feedback stabilizability in a nonuniform framework. Comptes Rendus. Mathématique, Tome 362 (2024) no. G12, pp. 1667-1692. doi: 10.5802/crmath.672

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