Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

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Dekhkonov F. N. On a time-optimal control problem for a heat conduction equation with involution. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2025, vol. 25, iss. 4, pp. 467-478. DOI: 10.18500/1816-9791-2025-25-4-467-478, EDN: GQNYBX

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
28.11.2025
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Language: 
English
Heading: 
Mathematics
Article type: 
Article
UDC: 
517.977
DOI: 
10.18500/1816-9791-2025-25-4-467-478
EDN: 
GQNYBX

On a time-optimal control problem for a heat conduction equation with involution

Autors: 
Dekhkonov Farrukh N., Namangan State University
Abstract: 

In this paper, we consider a boundary control problem for a heat conduction equation with involution in a bounded one-dimensional domain. The solution with the control function on the border of the rod is given. The constraints on the control are determined to ensure that the average value of the solution within the considered domain attains a given value. The considered control problem is reduced to the Volterra integral equation, which is the first type, using the Fourier method. The proof of the existence of admissible control is related to the existence of a solution of the integral equation. The existence of the control function was proved by the Laplace transform method, and the estimate of the minimum time to reach the given average temperature in the rod was found.

Key words: 
Initial-boundary problem
heat equation
minimal time
integral equation
admissible control
Laplace transform
involution
Acknowledgments: 
The author is grateful to Academician Sh. A. Alimov for his valuable comments.
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Received: 
03.10.2024
Accepted: 
12.02.2025
Published: 
28.11.2025
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