A block moment method to handle spectral condensation phenomenon in parabolic control problems

Benabdallah, Assia; Boyer, Franck; Morancey, Morgan

Annales Henri Lebesgue, Tome 3 (2020), pp. 717-793.

Cet article est consacré à la caractérisation du temps minimal de contrôle à zéro pour des problèmes de contrôles linéaires abstraits. Plus précisément, notre but est de répondre précisément à la question : quel est le temps minimal nécessaire pour amener à zéro la solution du problème issue de n’importe quelle condition initiale dans un sous-espace donné ? Notre cadre d’étude englobe un grand nombre de systèmes d’équations paraboliques linéaires unidimensionnelles couplées avec un contrôle scalaire.

Par une approche classique, ce problème de contrôle est ramené à la résolution d’un problème de moments dont nous proposons une nouvelle méthode de résolution par blocs. Les estimées obtenues lors de cette résolution sont optimales et uniformes pour certaines classes d’opérateurs. Cette uniformité a de nombreuses applications dans l’étude des problèmes de contrôle dépendants d’un paramètre et nous a permis de traiter simplement le cas où l’opérateur d’évolution possède des valeurs propres algébriquement multiples.

Cette approche nous a permis de mettre en lumière un nouveau phénomène : la condensation des valeurs propres (qui, en général, peut être une cause de temps minimal de contrôle à zéro strictement positif) peut être d’une certaine manière compensée par la condensation des fonctions propres. Pour illustrer cela et mettre en valeur la résolution par blocs, nous traitons différents exemples (aussi bien pour des systèmes abstraits que sur des problèmes d’EDPs).

This article is devoted to the characterization of the minimal null control time for abstract linear control problem. More precisely we aim at giving a precise answer to the following question: what is the minimal time needed to drive the solution of the system starting from any initial condition in a given subspace to zero? Our setting will encompass a wide variety of systems of coupled one dimensional linear parabolic equations with a scalar control.

Following classical ideas we reduce this controllability issue to the resolution of a moment problem on the control and provide a new block resolution technique for this moment problem. The obtained estimates are sharp and hold uniformly for a certain class of operators. This uniformity allows various applications for parameter dependent control problems and permits us to deal naturally with the case of algebraically multiple eigenvalues in the underlying generator.

Our approach sheds light on a new phenomenon: the condensation of eigenvalues (which can cause a non zero minimal null control time in general) can be somehow compensated by the condensation of eigenvectors. We provide various examples (some are abstract systems, others are actual PDE systems) to highlight those new situations we are able to manage by the block resolution of the moment problem we propose.

Classification : 93B05, 93C20, 93C25, 30E05, 35K90, 35P10
Mots clés : control theory, parabolic partial differential equations, minimal null control time, block moment method

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