matmessМатематические заметки СВФУMathematical notes of NEFU2411-93262587-876XСеверо-Восточный федеральный университет имени М.К. Аммосова10.25587/SVFU.2023.82.41.005matmess-178Research ArticleМАТЕМАТИКАON THE CONTROL PROBLEM ASSOCIATED WITH THE HEATING PROCESSON THE CONTROL PROBLEM ASSOCIATED WITH THE HEATING PROCESSDekhkonovF. N.DekhkonovF. N.

Farrukh N. Dekhkonov

National University of Uzbekistan

4, University street, Tashkent 100174, Uzbekistan

Farrukh N. Dekhkonov

National University of Uzbekistan

4, University street, Tashkent 100174, Uzbekistan

f.n.dehqonov@mail.ruNational University of UzbekistanУзбекистанNational University of UzbekistanUzbekistan2022250320262946372Copyright © Dekhkonov F.N., 20262026Dekhkonov F.N.Dekhkonov F.N.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://matmess.elpub.ru/jour/article/view/178

Abstract: In the papaer, we consider the initial-boundary problem for the heat conduction equation inside a bounded 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. It is supposed that on the boundary of this domain the heat exchange takes place according to Newton’s law. The control parameter is equal to the magnitude of output of hot or cold air and is defined on a given part of the boundary, and the weight function is not assumed to be strictly positive in the given domain. Then, we found the dependence T(θ) on the parameters of the temperature process when θ is close to critical value.

Abstract: In the papaer, we consider the initial-boundary problem for the heat conduction equation inside a bounded 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. It is supposed that on the boundary of this domain the heat exchange takes place according to Newton’s law. The control parameter is equal to the magnitude of output of hot or cold air and is defined on a given part of the boundary, and the weight function is not assumed to be strictly positive in the given domain. Then, we found the dependence T(θ) on the parameters of the temperature process when θ is close to critical value.

heat conduction equationcontrol functioninitial-boundary value problemadmissible controlintegral equation.heat conduction equationcontrol functioninitial-boundary value problemadmissible controlintegral equation.ReferencesTikhonov A. N. and Samarsky A. A., Equations of Mathematical Physics, Nauka, MoscowTikhonov A. N. and Samarsky A. A., Equations of Mathematical Physics, Nauka, Moscow(1966).(1966).Ladyzhenskaya O. A., Solonnikov V.A., and Uraltseva N. N., Linear and Quasi-Linear Equations of Parabolic Type [Russian], Nauka, Moscow (1967).Ladyzhenskaya O. A., Solonnikov V.A., and Uraltseva N. N., Linear and Quasi-Linear Equations of Parabolic Type [Russian], Nauka, Moscow (1967).Alimov Sh. A. and Dekhkonov F. N., "On the time-optimal control of the heat exchange process", Uzbek Mat. J., No. 2, 4–17 (2019).Alimov Sh. A. and Dekhkonov F. N., "On the time-optimal control of the heat exchange process", Uzbek Mat. J., No. 2, 4–17 (2019).Dekhkonov F. N., "On time-optimal control problem associated with parabolic equation", Bul. National Univ. Uzbekistan, 4, Iss. 1, Article 5 (2021).Dekhkonov F. N., "On time-optimal control problem associated with parabolic equation", Bul. National Univ. Uzbekistan, 4, Iss. 1, Article 5 (2021).Dekhkonov F. N., "On a time-optimal control of thermal processes in a boundary value problem", Lobachevskii L. Mat. 43, No. 1, 192–198 (2022).Dekhkonov F. N., "On a time-optimal control of thermal processes in a boundary value problem", Lobachevskii L. Mat. 43, No. 1, 192–198 (2022).Egorov Yu. V., "Optimal control in Banach spaces", Dokl. Akad. Nauk SSSR [in Russian] 150, 241–244 (1963).Egorov Yu. V., "Optimal control in Banach spaces", Dokl. Akad. Nauk SSSR [in Russian] 150, 241–244 (1963).Fattorini H. O., "Time-optimal control of solutions of operational differential equations", SIAM J. Control, 2, 49–65 (1964).Fattorini H. O., "Time-optimal control of solutions of operational differential equations", SIAM J. Control, 2, 49–65 (1964).Albeverio S. and Alimov Sh. A., "On one time-optimal control problem associated with the heat exchange process", Appl. Mat. Optimiz., 47, No. 1, 58–68 (2008).Albeverio S. and Alimov Sh. A., "On one time-optimal control problem associated with the heat exchange process", Appl. Mat. Optimiz., 47, No. 1, 58–68 (2008).Altmu¨ller A. and Gru¨ne L., Distributed and boundary model predictive control for the heat equation, Technical report, University of Bayreuth, Department of Mathematics, 2012.Altmu¨ller A. and Gru¨ne L., Distributed and boundary model predictive control for the heat equation, Technical report, University of Bayreuth, Department of Mathematics, 2012.Chen N., Wang Y., and Yang D., "Time–varying bang–bang property of time optimal controls for heat equation and its applications", Syst. Control Lett. 112, 18–23 (2018).Chen N., Wang Y., and Yang D., "Time–varying bang–bang property of time optimal controls for heat equation and its applications", Syst. Control Lett. 112, 18–23 (2018).Fattorini H. O. "Time and norm optimal controls: a survey of recent results and open problems", Acta Math. Sci. Ser. B Engl. Ed. 31, 2203–2218 (2011).Fattorini H. O. "Time and norm optimal controls: a survey of recent results and open problems", Acta Math. Sci. Ser. B Engl. Ed. 31, 2203–2218 (2011).Friedman A., Optimal control for parabolic equations", J. Mat. Anal. Appl. 18, 479–491Friedman A., Optimal control for parabolic equations", J. Mat. Anal. Appl. 18, 479–491(1967).(1967).Fursikov A. V. Optimal Control of Distributed Systems. Theory and Applications, Amer. Math. Soc., Providence, Rhode Island (2000) (Transl. Mat. Monogr., 187).Fursikov A. V. Optimal Control of Distributed Systems. Theory and Applications, Amer. Math. Soc., Providence, Rhode Island (2000) (Transl. Mat. Monogr., 187).Lions J. L. Contro´le optimal de syst`emes gouvern´es par des ´equations aux d´eriv´ees partielles, Dunod Gauthier-Villars, Paris (1968).Lions J. L. Contro´le optimal de syst`emes gouvern´es par des ´equations aux d´eriv´ees partielles, Dunod Gauthier-Villars, Paris (1968).Miranda C. Equazioni alle derivate parziali di tipo ellittico, Springer-Verl., Berlin; Go¨ttingen; Heidelberg (1955).Miranda C. Equazioni alle derivate parziali di tipo ellittico, Springer-Verl., Berlin; Go¨ttingen; Heidelberg (1955).Vladimirov V. S. Equations of Mathematical Physics, Marcel Dekker, New York (1971).Vladimirov V. S. Equations of Mathematical Physics, Marcel Dekker, New York (1971).

The authors declare that there are no conflicts of interest present.