The observability inequalities for heat equations with potentials

Document Type

Article

Publication Date

1-1-2025

Abstract

This paper is mainly concerned with the observability inequalities for heat equations with time-dependent Lipschitz 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||2/3) (even for time-independent potentials). In this paper, we show that, for Lipschtiz potentials, this factor can be replaced by exp(C(||V||1/2 + ||tV ||1/3)), which improves the previous bound exp(C|| V ||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 observability inequality with optimal constant on arbitrary measurable subsets of positive measure both in space and time.

Publication Source (Journal or Book title)

ESAIM Control Optimisation and Calculus of Variations

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