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2 results on '"integral thickness condition"'

1. APPROXIMATE NULL-CONTROLLABILITY WITH UNIFORM COST FOR THE HYPOELLIPTIC ORNSTEIN-UHLENBECK EQUATIONS.

Author

ALPHONSE, PAUL and MARTIN, JÉRÉMY

Subjects
*OPERATOR equations, *HEAT equation, *EQUATIONS, *CARLEMAN theorem, *ELLIPTIC equations, *COST
Abstract

We prove that the approximate null-controllability with uniform cost of the hypoelliptic Ornstein-Uhlenbeck equations posed on Rn is characterized by an integral thickness geometric condition on the control supports. We also provide associated quantitative weak observability estimates. This result for the hypoelliptic Ornstein-Uhlenbeck equations is deduced from the same study for a large class of nonautonomous elliptic equations from moving control supports. We generalize in particular results known for parabolic equations posed on Rn, for which the approximate null-controllability with uniform cost is ensured by the notion of thickness, which is stronger than the integral thickness condition considered in the present paper. Examples of those parabolic equations are the fractional heat equations associated with the operator (-Δ)s in the regime s ≥ 1/2. Our strategy also allows us to characterize the approximate null-controllability with uniform cost from moving control supports for this class of fractional heat equations. [ABSTRACT FROM AUTHOR]

Published
2023
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2. Approximate null-controllability with uniform cost for the hypoelliptic Ornstein-Uhlenbeck equations

Author

Alphonse, Paul, Martin, Jérémy, Unité de Mathématiques Pures et Appliquées (UMPA-ENSL), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Lyon (ENS Lyon), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), École normale supérieure de Lyon (ENS de Lyon)-Centre National de la Recherche Scientifique (CNRS), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), and Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest

Subjects
Mathematics - Analysis of PDEs, Optimization and Control (math.OC), unique continuation property, 93B05, 35H10, 47D06, 42B37, hypoellipticity, FOS: Mathematics, approximate null-controllability, Ornstein-Uhlenbeck equations, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], integral thickness condition, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Mathematics - Optimization and Control, Analysis of PDEs (math.AP)
Abstract

We prove that the approximate null-controllability with uniform cost of the hypoelliptic Ornstein-Uhlenbeck equations posed on $\mathbb R^n$ is characterized by an integral thickness geometric condition on the control supports. We also provide associated quantitative weak observability estimates. This result for the hypoelliptic Ornstein-Uhlenbeck equations is deduced from the same study for a large class of non-autonomous elliptic equations from moving control supports. We generalize in particular results known for parabolic equations posed on $\mathbb R^n$, for which the approximate null-controllability with uniform cost is ensured by the notion of thickness, which is stronger that the integral thickness condition considered in the present work. Examples of those parabolic equations are the fractional heat equations associated with the operator $(-\Delta)^s$, in the regime $s\geq1/2$. Our strategy also allows to characterize the approximate null-controllability with uniform cost from moving control supports for this class of fractional heat equations.

Published
2022
Full Text
View/download PDF

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