# 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

**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)

**References**

- Albeverio S., Alimov Sh. A.On a time-optimal control problem associated with the heat exchange process, Applied Mathematics and Optimization, 2008, vol. 47, no. 1, pp. 58–68.
- Alimov Sh. A.On a control problem associated with the heat transfer process, Eurasian Mathematical Journal, 2010. no. 1, pp. 17–30.
- Alimov Sh. A., Dekhkonov F. N.On a control problem associated with fast heating of a thin rod, Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences, 2019. vol. 2, no. 1, pp. 1–14.
- Alimov Sh. A., Dekhkonov F. N.On the time-optimal control of the heat exchange process, Uzbek Mathematical Journal, 2019. no. 2, pp. 4–17.
- Altmüller A., Grüne L. Distributed and boundary model predictive control for the heat equation, Technical report,. University of Bayreuth, Department of Mathematics, 2012.
- Chen N., Wang Y., Yang D. Time–varying bang–bang property of time optimal controls for heat equation and its applications, Syst. Control Lett, 2018. vol. 112, pp. 18–23.
- Egorov Yu. V. Optimal control in Banach spaces, Dokl. Akad. Nauk SSSR, 1963. vol. 150, pp. 241–244 (In Russian).
- Fattorini H. O. Time-optimal control of solutions of operational differential equations, SIAM J. Control, 1964. vol. 2, pp. 49–65.
- Fattorini H. O. Time and norm optimal controls: a survey of recent results and open problems,Acta Math. Sci. Ser. B Engl. Ed., 2011. vol. 31, pp. 2203–2218.
- Friedman A. Optimal control for parabolic equations, J. Math. Anal. Appl., 1967. vol. 18, pp. 479–491.
- Dekhkonov F. N.On a time-optimal control of thermal processes in a boundary value problem,Lobachevskii journal of mathematics, 2022. vol. 43, no. 1, pp. 192–198.
- Dekhkonov F. N.On time-optimal control problem associated with parabolic equation, Bulletin of National University of Uzbekistan, 2021. vol. 4, no. 1, pp. 54–63.
- Dekhkonov F. N.On the control problem associated with the heating process in the bounded domain, Vestnik KRAUNC. Fiz.-Mat. Nauki, 2022. vol. 39, no. 2, pp. 20–31.
- Fayazova Z. K. Boundary control of the heat transfer process in the space,Russian Mathematics (Izvestiya VUZ. Matematika), 2019. vol. 63, no. 12, pp. 71–79.
- Fayazova Z. K. Boundary control for a Psevdo-Parabolic equation, Mathematical notes of NEFU, 2018. vol. 25, no. 2, pp. 40–45.
- Il’in V. A., Moiseev E. I. Optimization of boundary controls of string vibrations, Rus. Math. Surveys, 2005. vol. 60, no. 6, pp. 1093–1119.
- Fursikov A. V. Optimal Control of Distributed Systems: Math. Soc., Prov., 2000.
- Ladyzhenskaya O. A., Solonnikov V. A., Uraltseva N. N. Linear and Quasi-Linear Equations of Parabolic Type. Moscow: Nauka, 1967 (In Russian).
- Lions J. L. Contróle optimal de systèmes gouvernés par deséquations aux dérivées partielles. Dunod Gauthier-Villars: Paris, 1968.
- Dubljevic S., Christofides P. D. Predictive control of parabolic PDEs with boundary control actuation. Chemical Engineering Science, 2006.
- Tikhonov A. N., Samarsky A. A. Equations of Mathematical Physics. Moscow, 1966.
- Vladimirov V. S. Equations of Mathematical Physics. Marcel Dekker: New York, 1971.

**Information about** **the author **

**Dekhkonov Farrukhjon Nuriddin ugli **– Ph.D. (Phys. & Math.), Associate Professor of the Department of Mathematical Analysis, Namangan State University, Namangan, Uzbekistan, https://orcid.org/0000-0003-4747-8557.