Boundary Control Problem for the Heat Equation with a Nonlocal Boundary Condition
Загрузки
In this paper, we study a boundary control problem for the one-dimensional heat
equation with a nonlocal boundary condition. The control is applied at one end of the domain, while
the temperature at the opposite boundary is linked to the temperature at a fixed interior point.
The control objective is formulated as an integral condition prescribing the average temperature of
the rod. Using the method of separation of variables, the problem is reduced to a Volterra integral
equation of the first kind. The existence of an admissible control function is proved by means of the
Laplace transform method.
1. Fattorini H.O., Time-optimal control of solutions of operational differential equations, SIAM J. Control,
2 (1965), pp. 49–65.
2. Friedman A., Optimal control for parabolic equations, J. Math. Anal. Appl., 18 (1967), pp. 479–491.
3. Egorov Yu.V., Optimal control in Banach spaces, Dokl. Akad. Nauk SSSR, 150 (1967), pp. 241–244. (in
Russian)
4. Albeverio S., Alimov Sh. A., On one time-optimal control problem associated with the heat exchange
process, Applied Mathematics and Optimization, 57 (2008), pp. 58–68.
5. Alimov Sh.A., Dekhkonov F.N., On the time-optimal control of the heat exchange process, Uzbek
Mathematical Journal, 2 (2019), pp. 4–17.
6. Alimov Sh.A., On the null-controllability of the heat exchange process, Eurasian Mathematical Journal,
2 (2011), pp. 5–19.
7. Dekhkonov F.N., On the control problem associated with the heating process, Mathematical notes of
NEFU, 29 (2022), pp. 62–71.
8. 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, 2 (2019), pp. 1–14.
9. 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 (2018), pp. 18–23.
10. Fayazova Z.K., Boundary control of the heat transfer process in the space, Izv. Vyssh. Uchebe. Zaved.
Mat., 12 (2019), pp. 82–90.
11. Dekhkonov F.N., Boundary control problem for the heat transfer equation associated with heating process
of a rod, Bulletin of the Karaganda University. Mathematics Series, 110 (2023), pp. 63–71.
12. Lions J.L., Contr´ ole optimal de syst` emes gouvern´ es par des´ equations aux d´ eriv´ ees partielles, Dunod
Gauthier-Villars, Paris, 1968.
13. Fursikov A.V., Optimal Control of Distributed Systems, Theory and Applications, Translations of Math.
Monographs, 187, Amer. Math. Soc., Providence, Rhode Island, 2000.
14. Altmüller A., Grüne L., Distributed and boundary model predictive control for the heat equation, Technical
report, University of Bayreuth, Department of Mathematics, 2012.
15. Dubljevic S., Christofides P.D., Predictive control of parabolic PDEs with boundary control actuation,
Chemical Engineering Science, 61 (2006), pp. 6239–6248.
16. Ashurov R. R., Kadirkulov B. J., Turmetov B. Kh., On the inverse problem of the Bitsadze-Samarskii
type for a fractional parabolic equation, Probl. Anal. Issues Anal., 2023, Issue 3, 20-40
17. Tikhonov A.N. and Samarsky A.A., Equations of Mathematical Physics, Nauka, Moscow. 1966.
18. Ladyzhenskaya O.A., Solonnikov V.A. and Uraltseva N.N., Linear and quasi-linear equations of parabolic
type, Nauka, Moscow. 1967.
Copyright (c) 2026 «ВЕСТНИК НУУз»
Это произведение доступно по лицензии Creative Commons «Attribution-NonCommercial-ShareAlike» («Атрибуция — Некоммерческое использование — На тех же условиях») 4.0 Всемирная.
.jpg)
2.png)
