# Vestnik КRAUNC. Fiz.-Mat. nauki. 2024. vol. 47. no. 2. P. 9 – 20. ISSN 2079-6641

**MATHEMATICS**

https://doi.org/10.26117/2079-6641-2024-47-2-9-20

Research Article

Full text in English

MSC 35K05, 35K15

**The Control Problem for a Heat Conduction Equation with Neumann Boundary Condition**

**F. N. Dekhkonov^{\ast}**

Namangan State University, 316, Uychi str., 160136, Namangan, Uzbekistan

Abstract. Previously, boundary control problems for a heat conduction equation with Dirichlet boundary condition were studied in a bounded domain. In this paper, we consider the boundary control problem for the heat conduction equation with Neumann boundary condition in a bounded one-dimensional domain. 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 studied initial boundary value problem is reduced to the Volterra integral equation of the first type using the method of separation of variables. It is known that the solution of Volterra’s integral equation of the first kind cannot always be shown to exist. In our work, the existence of a solution to the Volterra integral equation of the first kind is shown using the method of Laplace transform. For this, the necessary estimates for the kernel of the integral equation were found. Finally, the admissibility of the control function is proved.

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

Received: 18.04.2024; Revised: 17.05.2024; Accepted: 08.06.2024; First online: 25.08.2024

**For citation.** Dekhkonov F. N. The control problem for a heat conduction equation with Neumann boundary condition. Vestnik KRAUNC. Fiz.-mat. nauki. 2024, 47: 2, 9-20. EDN: MNMAFB. https://doi.org/10.26117/2079-6641-2024-47-2-9-20.

**Funding.** The work was not carried out within the framework of funds

**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 the press.

^{\ast}**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., 2024

© Institute of Cosmophysical Research and Radio Wave Propagation, 2024 (original layout, design, compilation)

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**Information about the author**

**Dekhkonov Farrukhjon Nuriddin ogli **– Ph.D. (Phys. & Math.), Namangan State University, Namangan, Uzbekistan, ORCID 0000-0003-4747-8557.