On the time-optimal control problem for a pseudo-parabolic equation with variable thermal conductivity in a bounded domain

Farrukh Dekhkonov; Zhuang Han; Malika Nematjonova; Weipeng Wu; Farrukh Dekhkonov; Zhuang Han; Malika Nematjonova; Weipeng Wu

doi:10.3934/cam.2026030

Communications in Analysis and Mechanics

2026, Volume 18, Issue 3: 732-748. doi: 10.3934/cam.2026030

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Research article

On the time-optimal control problem for a pseudo-parabolic equation with variable thermal conductivity in a bounded domain

1.
Department of Mathematics, Namangan State University, Namangan 160136, Uzbekistan
2.
College of Mathematical Sciences, Harbin Engineering University, Harbin 150001, China
3.
Harbin Institute of Information Technology, Harbin, Heilongjiang 150431, China

Received: 05 April 2026 Revised: 27 May 2026 Accepted: 03 July 2026 Published: 15 September 2026
35K70, 35K35

This paper studied a time-optimal control problem for a pseudo-parabolic equation in a bounded multidimensional domain. The equation involved a uniformly elliptic operator and a small parameter characterizing memory effects. The control acted through a distributed source term, where the spatial part was fixed and the time-dependent part was a bounded measurable function. The goal was to steer the system from a zero initial state to a prescribed spatially averaged temperature in minimal time and maintain it thereafter. Using spectral decomposition, we derived an explicit solution representation and reduced the control problem to a Volterra integral equation. Under natural positivity assumptions on the Fourier coefficients, we established that the kernel was continuous, positive, and monotone. These properties yielded an explicit upper bound for the minimal time in terms of the first eigenvalue, the relaxation parameter, and the system constraints. A comparison with the parabolic case showed that the pseudo-parabolic term led to longer minimal times, reflecting the memory effects inherent in such models.
- pseudo-parabolic equation,
- thermal conductivity,
- Volterra integral equation,
- admissible control,
- minimal time,
- memory effect
Citation: Farrukh Dekhkonov, Zhuang Han, Malika Nematjonova, Weipeng Wu. On the time-optimal control problem for a pseudo-parabolic equation with variable thermal conductivity in a bounded domain[J]. Communications in Analysis and Mechanics, 2026, 18(3): 732-748. doi: 10.3934/cam.2026030

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Abstract

This paper studied a time-optimal control problem for a pseudo-parabolic equation in a bounded multidimensional domain. The equation involved a uniformly elliptic operator and a small parameter characterizing memory effects. The control acted through a distributed source term, where the spatial part was fixed and the time-dependent part was a bounded measurable function. The goal was to steer the system from a zero initial state to a prescribed spatially averaged temperature in minimal time and maintain it thereafter. Using spectral decomposition, we derived an explicit solution representation and reduced the control problem to a Volterra integral equation. Under natural positivity assumptions on the Fourier coefficients, we established that the kernel was continuous, positive, and monotone. These properties yielded an explicit upper bound for the minimal time in terms of the first eigenvalue, the relaxation parameter, and the system constraints. A comparison with the parabolic case showed that the pseudo-parabolic term led to longer minimal times, reflecting the memory effects inherent in such models.

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