Observability from Measurable Sets for One-Dimensional Heat Equations with Bounded Spacetime Potentials

Xiaomin Zhu

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arXiv · 2610.05704

Observability from Measurable Sets for One-Dimensional Heat Equations with Bounded Spacetime Potentials

Xiaomin Zhu

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Abstract

This paper study observability for the Dirichlet heat equation on a bounded interval with a real bounded potential depending on space and time. We prove an observability inequality from every measurable subset of spacetime with positive measure. The constant is uniform over potentials with a prescribed $L^\infty$ bound. By duality, positive spacetime measure is equivalent to null controllability by square-integrable distributed controls. Controls can also be chosen bounded in time with values in $L^2$. If almost every spatial slice of the observation set has a fixed positive lower bound on its measure, the observability constant is bounded above by $C_0e^{C_1/T}$, uniformly in the locations of the slices. The proof combines observation estimates on finite-dimensional spaces transported by the evolution with decay estimates for the distance to those spaces.

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Xiaomin Zhu. 2026-10-05. Observability from Measurable Sets for One-Dimensional Heat Equations with Bounded Spacetime Potentials. https://arxiv.org/abs/2610.05704

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