Vestnik КRAUNC. Fiz.-Mat. Nauki. 2023. vol. 42. no. 1. P. 69-79. ISSN 2079-6641

MATHEMATICS
https://doi.org/10.26117/2079-6641-2023-42-1-69-79
Research Article
Full text in English
MSC 35K05, 35K15

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Control Problem Concerned With the Process of Heating a Thin Plate

F. N. Dekhkonov^*

Namangan State University, Uzbekistan, 160136, Namangan B. Mashrab, 1A.

Abstract. Previously, a mathematical model for the following problem was considered. On a part of the border of the right rectangle there is a heater with controlled temperature. It is required to find such a mode of its operation that the average temperature in some region reaches some given value. In this paper, we consider a boundary control problem associated with a parabolic equation on a right rectangle. On the part of the border of the considered domain, the value of the solution with control parameter is given. Restrictions on the control are given in such a way that the average value of the solution in some part of the considered domain gets a given value. The auxiliary problem is solved by the method of separation of variables, while the problem in consideration is reduced to the Volterra integral equation. In addition, the definition of the generalized solution of the given initialboundary problem is given in the article and the existence of such a solution is proved. The solution of Volterra’s integral equation was found by the Laplace transform method and the existence theorem for admissible control functions was proved. It is also shown that the initial value of the admissible control function is equal to zero using the change of variable in the integral equation. The proof of this comes from the fact that the kernels of the integral equations are positive and finite, and the system has a single-valued solution.

Key words: parabolic equation, system of integral equations, initial-boundary problem, admissible control, Laplace transform.

Received: 11.01.2023; Revised: 14.03.2023; Accepted: 20.03.2023; First online: 15.04.2023

For citation. Dekhkonov F. N. Control problem concerned with the process of heating a thin plate. Vestnik KRAUNC.
Fiz.-mat. nauki. 2023, 42: 1, 69-79. EDN: DJZRAU. https://doi.org/10.26117/2079-6641-2023-42-1-69-79.

Funding. Not applicable.

Competing interests. There are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. The author participated in the writing of the article and is fully responsible for
submitting the final version of the article to print.

^*Correspondence: E-mail: f.n.dehqonov@mail.ru

The content is published under the terms of the Creative Commons Attribution 4.0 International License

© Dekhkonov F. N., 2023

© Institute of Cosmophysical Research and Radio Wave Propagation, 2023 (original layout, design, compilation)

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