2024/09/14 by Jiuyi Zhu, Zhu, Jiuyi, Jinping Zhuge +1 · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Optimization and Control (math.OC) #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2409.09476
openalex publication_date 2024/09/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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‖∞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 optimal observability constant.