In this work, we consider boundary control problem associated with a parabolic equation on a interval. On the part of the border of the considered segment, 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 interval 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 of the second kind. The control parameter is defined on one. The estimate of a minimal time for achieving the given average and at the interval temperature is found.
1. Alimov Sh.A. On a control problem associated with the heat transfer process. Eurasian Mathematical Journal, Vol. 1, No. 2, 17–30 (2010).
2. Albeverio S., Alimov Sh.A. On one time-optimal control problem associated with the heat exchange process. Applied Mathematics and Optimization, Vol. 47, No. 1, 58–68 (2008).
3. Alimov Sh.A., Dekhkonov F.N. On the time-optimal control of the heat exchange process. Uzbek Mathematical Journal, No. 2, 4–17 (2019).
4. Alimov Sh.A. On the null-controllability of the heat exchange process. Eurasian Mathematical Journal, No. 2, 5–19 (2011).
5. Alimov Sh.A., Umarov A.T. On the null-controllability of the heat exchange process. Acta NUUz, No. 3, 21–24 (2010).
6. Fattorini H.O. Time-Optimal control of solutions of operational differential equations. SIAM J. Control, Vol. 2, Issue 1, 54–59 (1964).
7. Fattorini H.O. Time and norm optimal control for linear parabolic equations: necessary and sufficient conditions. Control and Estimation of Distributed Parameter Systems. ISNM International Series of Numerical Mathematics, Vol. 143, Birkhaüser, Besel, 151–168 (2003).
8. Barbu V. The time-optimal control problem for parabolic variotianal inequalities. Applied Mathematics and Optimization, Vol. 11, Issue 1, 1–22 (1984).
9. Friedman A. Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, XVI (1964).
10. Fursikov A.V. Optimal Control of Distributed Systems, Theory and Applications, Translations of Math. Monographs, Vol. 187, Amer. Math. Soc., Providence, Rhode Island. (2000)
11. Altmüller A., Grüne L. Distributed and boundary model predictive control for the heat equation. Technical report, University of Bayreuth, Department of Mathematics (2012).
12. Dubljevic S., Christofides P.D. Predictive control of parabolic PDEs with boundary control actuation. Chemical Engineering Science, Vol. 61, Issue 18, 6239–6248 (2006).
13. Lions J.L. Contròle optimal de systèmes gouvernés par des équations aux dérivées partielles. Dunod Gauthier-Villars, Paris (1968).
14. Il'in V.A. Boundary control by the oscillation process at two ends in terms of the generalized solution of the wave equation with finite energy. Differ. Equ., Vol. 36, No. 11, 1659–1675 (2000).
15. Satimov N.Yu., Tukhtasinov M. Game problems on a fixed interval in controlled first-order evolution equations. Mathematical Notes, Vol. 80, No. 4, 578–589 (2006).
16. Tukhtasinov M., Ibragimov U. Sets invariant under an integral constraint on controls. Russian Math. (Iz. VUZ), Vol. 55, No. 8, 59–65 (2011).
17. Fayazova Z.K. Boundary control for a Psevdo-Parabolic equation. Mathematical notes of NEFU, Vol. 25, No. 2, 40–45 (2018).
18. Tukhtasinov M., Mustapokulov Kh., Ibragimov G. Invariant Constant Multi-Valued Mapping for the Heat Conductivity Proplem. Malasian Journal of Mathematical Sciences, Vol. 13, No. 1, 61–74 (2019).
19. Butkovsky A.G. Theory of Optimal of Distributed Parametr Systems. Elsevier, New York (1969).
20. Tikhonov A.N., Samarsky A.A. Equations of Mathematical Physics. Nauka, Moscow (1966).
21. Vladimirov V.S. Equations of Mathematical Physics. Marcel Dekker, New York (1971).
Alimov, Shavkat and Dekhkonov, Farrukh
"On a control problem associated with fast heating of a thin rod,"
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