Observability inequalities for heat equations with potentials

TL;DR

This paper improves the bounds on observability inequalities for heat equations with Lipschitz potentials, leading to better control estimates and optimal constants in specific cases.

Contribution

It introduces a refined observability inequality for heat equations with Lipschitz potentials, replacing previous bounds with sharper estimates involving derivatives of the potential.

Findings

Improved observability constant bounds for Lipschitz potentials.

Enhanced null controllability results using the new bounds.

Established optimal observability constants for 1D heat equations with bounded potentials.

Abstract

This paper is mainly concerned with the observability inequalities for heat equations with time-dependent Lipschtiz potentials. The observability inequality for heat equations asserts that the total energy of a solution is bounded above by the energy localized in a subdomain with an observability constant. For a bounded measurable potential $V = V(x,t)$, the factor in the observability constant arising from the Carleman estimate is best known to be $\exp(C\|V\|_{\infty}^{2/3})$ (even for time-independent potentials). In this paper, we show that, for Lipschtiz potentials, this factor can be replaced by $\exp(C(\|\nabla V\|_{\infty}^{1/2} +\|\partial_tV\|_{\infty}^{1/3} ))$, which improves the previous bound $\exp(C\|V\|_{\infty}^{2/3})$ in some typical scenarios. As a consequence, with such a Lipschitz potential, we obtain a quantitative regular control in a null controllability problem. In addition, for the one-dimensional heat equation with some time-independent bounded measurable potential $V = V(x)$, we obtain the optimal observability constant.