Control of Parabolic Equations with Inverse Square Infinite Potential Wells

This survey summarises a presentation recently given by the author at the Ghent Methusalem Junior Analysis Seminar. The talk discussed the recent result of Enciso et al. (Controllability of parabolic equations with inverse square infinite potential wells via global Carleman estimates. Preprint, 2021), joint with Alberto Enciso (ICMAT) and Bruno Vergara (Brown), as well as the main ideas of its proof. In Enciso et al. (Controllability of parabolic equations with inverse square infinite potential wells via global Carleman estimates. Preprint, 2021), we consider heat operators on a bounded convex domain, with a critically singular potential diverging as the inverse square of the distance to the boundary of the domain. We address the problem of boundary null controllability—whether one can drive the solution from any initial data to zero via suitable boundary data. We establish a null control result for such operators in all spatial dimensions, in particular providing the first result in more than one spatial dimension. The key step in the proof is a novel global Carleman estimate that captures both the relevant boundary asymptotics and the appropriate energy for this problem.