Approximate boundary controllability for parabolic equations with inverse square infinite potential wells

Abstract

We consider heat operators on a bounded domain Ω⊆Rn, with a critically singular potential diverging as the inverse square of the distance to ∂Ω. Although null boundary controllability for such operators was recently proved in all dimensions in Enciso et al. (2023) , it crucially assumed (i) Ω was convex, (ii) the control must be prescribed along all of ∂Ω, and (iii) the strength of the singular potential must be restricted to a particular subrange. In this article, we prove instead a definitive approximate boundary control result for these operators, in that we (i) do not assume convexity of Ω, (ii) allow for the control to be localized near any x0∈∂Ω, and (iii) treat the full range of strength parameters for the singular potential. Moreover, we lower the regularity required for ∂Ω and the lower-order coefficients. The key novelty is a local Carleman estimate near x0, with a carefully chosen weight that takes into account both the appropriate boundary conditions and the local geometry of ∂Ω.