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

Abstract

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 and observed from a non-empty open subset More precisely, for our main result shows that, when has a finite number of holes, the observability constant of the elliptic operator with domain is of the form 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 [1], 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 compared to [1] 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.