Boundary null controllability of some multi-dimensional linear parabolic systems by the moment method
[Contrôlabilité à zéro par le bord de certains systèmes paraboliques linéaires multi-dimensionnels par la méthode des moments]
Boyer, Franck1 ; Olive, Guillaume2
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 Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Łojasiewicza 6, 30-348 Kraków (Poland)
Annales de l'Institut Fourier, Online first, 70 p.

Dans cet article nous étudions la contrôlabilité à zéro d’une classe de systèmes paraboliques linéaires abstraits dans des produits tensoriels. Cette structure particulière nous permet de réduire la question de la contrôlabilité à un ensemble particulier d’équations qui ressemble à un problème de moments, mais qui ne relève pas des résultats existants de la littérature.

Nous transformons ce problème de moments non standard en une famille infinie de problèmes de moments usuels, mais couplés entre eux. Cette reformulation est choisie avec soin pour que le nouveau système obtenu puisse être résolu, avec des bonnes estimations des solutions, en utilisant la méthode des moments par blocs développée récemment. Tout ce travail est basé sur une analyse spectrale minutieuse de l’opérateur sous-jacent.

Nous appliquons notamment ce résultat abstrait pour montrer que la position du domaine de contrôle pour un problème de contrôle au bord de deux équations de la chaleur couplées peut être déterminante : nous donnons un exemple explicite d’un tel système posé sur un domaine rectangulaire en 2D qui est contrôlable à zéro en tout temps si le contrôle agit sur deux bords perpendiculaires du domaine mais qui n’est jamais contrôlable à zéro si le contrôle n’agit que sur un seul des bords du domaine.

In this article we study the null controllability of some abstract linear parabolic systems in tensor product spaces. This special structure allows us to reduce our controllability problem to a particular set of equations that looks like a moment problem, but that does not fall into the previous existing results of the literature.

We transform this non standard moment problem into an infinite family of more usual moment problems, yet coupled one from each other. This reformulation is done with enough care to ensure that the resulting set of equations can be solved, with suitable estimates, by using the recent “block moment method”. This is based on a careful analysis of the spectral structure of the underlying operator.

We notably apply our abstract result to show how strong the influence of geometry can be: we provide an example of boundary controlled parabolic system on a rectangle domain which is null controllable in arbitrary small time if two perpendicular faces of the boundary are controlled, whereas it is never null controllable if the control acts on only one face.

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DOI : 10.5802/aif.3639
Classification : 93B05, 93C20, 93C25, 30E05, 35K90, 47A80
Keywords: Controllability, Parabolic systems, Geometric control conditions, Block moment method, Tensor products.
Mot clés : Contrôlabilité, Systèmes paraboliques, Conditions de contrôle géométrique, Méthode des moments par blocs, Produits tensoriels.
Boyer, Franck 1 ; Olive, Guillaume 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 Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Łojasiewicza 6, 30-348 Kraków (Poland) 
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Boyer, Franck; Olive, Guillaume. Boundary null controllability of some multi-dimensional linear parabolic systems by the moment method. Annales de l'Institut Fourier, Online first, 70 p.

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