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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∞ 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 L2. 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 C0eC1/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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Cite

BibTeXRIS
BibTeX
@misc{zhu2026observability,
  title = {{Observability from Measurable Sets for One-Dimensional Heat Equations with Bounded Spacetime Potentials}},
  author = {Xiaomin Zhu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.05704},
  eprint = {2610.05704},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.05704},
}
RIS
TY  - UNPB
TI  - Observability from Measurable Sets for One-Dimensional Heat Equations with Bounded Spacetime Potentials
AU  - Xiaomin Zhu
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.05704
UR  - https://arxiv.org/abs/2610.05704
AB  - 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^∞ 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.
ER  -