Time-optimal control problem for a heat conduction equation with variable thermal conductivity

Mousa J. Huntul; Farrukh Dekhkonov; Malika Nematjonova; Mousa J. Huntul; Farrukh Dekhkonov; Malika Nematjonova

doi:10.3934/math.2026806

AIMS Mathematics

2026, Volume 11, Issue 7: 19855-19875. doi: 10.3934/math.2026806

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Research article

Time-optimal control problem for a heat conduction equation with variable thermal conductivity

1.
Department of Mathematics, College of Science, Jazan University, P.O. Box. 114, Jazan 45142, Saudi Arabia
2.
Department of Mathematics, Namangan State University, Namangan, Uzbekistan
3.
V.I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Science, Tashkent, Uzbekistan

Received: 15 February 2026 Revised: 30 June 2026 Accepted: 02 July 2026 Published: 07 July 2026
MSC : 35K05, 35K15

This paper investigates a time-optimal boundary control problem for the heat equation governed by a uniformly elliptic operator in a bounded multidimensional domain. Robin boundary conditions model passive heat exchange, while the control acts through a prescribed heat flux on a portion of the boundary. The admissible controls are assumed to be bounded measurable functions. The analysis relies on the spectral properties of the associated elliptic operator, including the discreteness of the spectrum and the positivity of the first eigenfunction. These properties are used to derive an optimal estimate for the minimal time required to achieve a prescribed average temperature in the domain.
- heat conduction equation,
- thermal conductivity,
- Volterra integral equation,
- admissible control,
- minimal time
Citation: Mousa J. Huntul, Farrukh Dekhkonov, Malika Nematjonova. Time-optimal control problem for a heat conduction equation with variable thermal conductivity[J]. AIMS Mathematics, 2026, 11(7): 19855-19875. doi: 10.3934/math.2026806

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Abstract

This paper investigates a time-optimal boundary control problem for the heat equation governed by a uniformly elliptic operator in a bounded multidimensional domain. Robin boundary conditions model passive heat exchange, while the control acts through a prescribed heat flux on a portion of the boundary. The admissible controls are assumed to be bounded measurable functions. The analysis relies on the spectral properties of the associated elliptic operator, including the discreteness of the spectrum and the positivity of the first eigenfunction. These properties are used to derive an optimal estimate for the minimal time required to achieve a prescribed average temperature in the domain.

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