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Journal Articles Communications in Partial Differential Equations Year : 2023

Cost of observability inequalities for elliptic equations in 2-d with potentials and applications to control theory

Sylvain Ervedoza

Abstract

The goal of this article is to obtain observability estimates for non-homogeneous elliptic equations in the presence of a potential, posed on a smooth bounded domain Ω in 2-d and observed from a non-empty open subset ω ⊂ Ω. More precisely, for every real-valued bounded potential V, our main result shows that, when Ω has a finite number of holes, the observability constant of the elliptic operator −∆ + V, with domain H^2 ∩ H^1_0(Ω), is of the form C exp (C |V|^{1/2} \log^{1/2}(|V|)) where C is a positive constant depending only on Ω and ω. Our methodology of proof is crucially based on the one recently developed by Logunov, Malinnikova, Nadirashvili, and Nazarov, in the context of the Landis conjecture on exponential decay of solutions to homogeneous elliptic equations in the plane. The main difference and additional difficulty is that the zero set of the solutions to elliptic equations with source term can be very intricate and should be dealt with carefully. As a consequence of these new observability estimates, we obtain new results concerning control of semi-linear elliptic equations in the spirit of Fernández-Cara, Zuazua's open problem concerning small-time global null-controllability of slightly super-linear heat equations.
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Year Month Jours
Avant la publication
Thursday, April 11, 2024
Embargoed file
Thursday, April 11, 2024
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Dates and versions

hal-03616317 , version 1 (22-03-2022)
hal-03616317 , version 2 (15-11-2023)

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Sylvain Ervedoza, Kévin Le Balc’h. Cost of observability inequalities for elliptic equations in 2-d with potentials and applications to control theory. Communications in Partial Differential Equations, 2023, 48 (4), pp.623--677. ⟨10.1080/03605302.2023.2190526⟩. ⟨hal-03616317v2⟩
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