On the time-optimal control problem for a fourth order parabolic equation in the three-dimensional space

Farrukh Dekhkonov; Wenke Li; Weipeng Wu; Farrukh Dekhkonov; Wenke Li; Weipeng Wu

doi:10.3934/cam.2025017

Communications in Analysis and Mechanics

2025, Volume 17, Issue 2: 413-428. doi: 10.3934/cam.2025017

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Research article

On the time-optimal control problem for a fourth order parabolic equation in the three-dimensional space

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

Received: 15 December 2024 Revised: 07 March 2025 Accepted: 18 March 2025 Published: 27 April 2025
Primary: 35K15, 35K35; Secondary: 58J35

In this paper, we consider the problem of optimal time control for a fourth-order parabolic-type equation describing the growth process of a thin film in a bounded three-dimensional space. The control function is defined on a certain part of a boundary. It is proved that the optimal time depends on the parameters of the growth process when the average value of the growth interface height of the thin film in the domain is close to the critical value.
- fourth order parabolic equation,
- Volterra integral equation,
- admissible control,
- initial-boundary problem,
- critical value,
- minimal time
Citation: Farrukh Dekhkonov, Wenke Li, Weipeng Wu. On the time-optimal control problem for a fourth order parabolic equation in the three-dimensional space[J]. Communications in Analysis and Mechanics, 2025, 17(2): 413-428. doi: 10.3934/cam.2025017

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Abstract

In this paper, we consider the problem of optimal time control for a fourth-order parabolic-type equation describing the growth process of a thin film in a bounded three-dimensional space. The control function is defined on a certain part of a boundary. It is proved that the optimal time depends on the parameters of the growth process when the average value of the growth interface height of the thin film in the domain is close to the critical value.

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