Analysis of non scalar control problems for parabolic systems by the block moment method

Franck Boyer; Morgan Morancey

doi:10.5802/crmath.487

Équations aux dérivées partielles, Théorie du contrôle

Analysis of non scalar control problems for parabolic systems by the block moment method
[Analyse de problèmes de contrôle non scalaires pour des systèmes paraboliques par la méthode des moments par blocs]

Franck Boyer ¹ ; Morgan Morancey ²

¹ Institut de Mathématiques de Toulouse & Institut Universitaire de France, UMR 5219, Université de Toulouse, CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France.
² Aix-Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1191-1248.

Résumé
Abstract

On étudie dans cet article des systèmes de contrôle paraboliques autonomes linéaires abstraits. Dans [9], avec A. Benabdallah, nous avons introduit la méthode des moments par blocs dans le cas d’un opérateur de contrôle scalaire. Le but principal de cette méthode est de permettre de calculer le temps minimal nécessaire pour amener à zéro une donnée initiale fixée (ou un espace de données initiales), en particulier dans le cas où des phénomènes de condensation spectrale sont présents. Le but du présent travail est d’approfondir cette analyse pour prendre en compte n’importe quel opérateur de contrôle admissible. Le cadre proposé permet des applications à des équations ou systèmes paraboliques couplés en dimension un d’espace.

Pour de tels opérateurs de contrôle admissibles, la caractérisation du temps minimal de contrôle est obtenu à l’aide de la résolution de problèmes de moment vectoriels auxiliaires suivie d’une procédure d’optimisation sous contrainte du coût de cette résolution. Cela amène à des estimations essentiellement optimales pour la résolution de ces problèmes de moment par bloc qui, de surcroît, sont uniformes par rapport au spectre de l’opérateur d’évolution à l’intérieur d’une certaine classe. Ce caractère uniforme permet de prouver la contrôlabilité uniforme de divers systèmes dépendant de paramètres. Nous déduisons également des estimations du coût de contrôlabilité quand le temps de contrôle est proche du temps minimal.

Nous illustrons le fonctionnement de cette méthode sur quelques exemples de tels systèmes abstraits mais également sur des exemples plus concrets de systèmes d’équations aux dérivées partielles paraboliques contrôlés en dimension 1. Notre stratégie permet d’étudier des propriétés de contrôlabilité qui semblent hors de portée par les méthodes existantes de la littérature.

This article deals with abstract linear time invariant controlled systems of parabolic type. In [9], with A. Benabdallah, we introduced the block moment method for scalar control operators. The principal aim of this method is to compute the minimal time needed to drive an initial condition (or a space of initial conditions) to zero, in particular in the case when spectral condensation occurs. The purpose of the present article is to push forward the analysis to deal with any admissible control operator. The considered setting leads to applications to one dimensional parabolic-type equations or coupled systems of such equations.

With such admissible control operator, the characterization of the minimal null control time is obtained thanks to the resolution of an auxiliary vectorial block moment problem (i.e. set in the control space) followed by a constrained optimization procedure of the cost of this resolution. This leads to essentially sharp estimates on the resolution of the block moment problems which are uniform with respect to the spectrum of the evolution operator in a certain class. This uniformity allows the study of uniform controllability for various parameter dependent problems. We also deduce estimates on the cost of controllability when the final time goes to the minimal null control time.

We illustrate how the method works on a few examples of such abstract controlled systems and then we deal with actual coupled systems of one dimensional parabolic partial differential equations. Our strategy enables us to tackle controllability issues that seem out of reach by existing techniques.

Reçu le : 2023-01-05
Accepté le : 2023-03-05
Publié le : 2023-10-31

DOI : 10.5802/crmath.487

Classification : 93B05, 93C20, 93C25, 30E05, 35K90, 35P10

Affiliations des auteurs :

Franck Boyer ¹ ; Morgan Morancey ²

¹ Institut de Mathématiques de Toulouse & Institut Universitaire de France, UMR 5219, Université de Toulouse, CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France.
² Aix-Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Analysis of non scalar control problems for parabolic systems by the block moment method},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1191--1248},
     publisher = {Acad\'emie des sciences, Paris},
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     language = {en},
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Franck Boyer; Morgan Morancey. Analysis of non scalar control problems for parabolic systems by the block moment method. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1191-1248. doi : 10.5802/crmath.487. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.487/

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