Vol. 25, no. 1, 2019
A control problem is considered for the process of heating a rod by varying the temperature at its left end. The exact values of the temperature at the right end of the rod and the heat density function are unknown; only the ranges of their possible values are given. The aim of the control is to ensure that the average temperature of the rod at a fixed time belongs to a given interval. We find necessary and sufficient conditions on the initial temperature of the rod under which the aim of the control can be achieved for any admissible unknown functions. The corresponding heating control at the left end of the rod is constructed.
Keywords: heat equation, temperature, control
Received December 10, 2018
Revised December 28, 2018
Accepted January 14, 2019
Funding Agency: This work was supported by the Russian Foundation for Basic Research (projects no. 18-01-00264_а).
Viktor Ivanovich Ukhobotov, Dr. Phys.-Math. Sci., Prof., Chelyabinsk State University, Chelyabinsk, 454001 Russia, e-mail: ukh@csu.ru
Igor’ Vyacheslavovich Izmest’ev, Cand. Sci. (Phys.-Math.), Chelyabinsk State University, Chelyabinsk, 454001 Russia, e-mail: j748e8@gmail.com
Cite this article as: V.I. Ukhobotov, I.V. Izmest’ev. A control problem for a rod heating process with unknown temperature at the right end and unknown density of the heat source, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 1, pp. 297-305.
REFERENCES
1. Osipov Yu.S. Position control in parabolic systems. J. Appl. Math. Mech., 1977, vol. 41, no. 2, pp. 187–193. doi: 10.1016/0021-8928(77)90001-6
2. Korotkii A.I., Osipov Yu.S. Approximation in problems of position control of parabolic systems. J. Appl. Math. Mech., 1978, vol. 42, no. 4, pp. 631–637. doi: 10.1016/0021-8928(78)90004-7
3. Egorov A.I. Optimal’noe upravlenie teplovymi i diffuzionnymi protsessami [Optimal control of thermal and diffusion processes]. Moscow: Nauka Publ., 1978, 464 p.
4. Vasil’ev F.P. Metody resheniya ekstremal’nykh zadach [Methods for solving extremal problems]. Moscow: Nauka Publ., 1981, 400 p.
5. Liu J., Zheng G., Ali M.M. Stability analysis of the anti-stable heat equation with uncertain disturbance on the boundary. J. Math. Anal. Appl., 2015, vol. 428, no. 2, pp. 1193–1201. doi: 10.1016/j.jmaa.2015.03.073
6. Dai J., Ren B. UDE-based robust boundary control of heat equation with unknown input disturbance. IFAC PapersOnLine, 2017, vol. 50, no. 1, pp. 11403–11408. doi: 10.1016/j.ifacol.2017.08.1801
7. Krasovskii N.N. Upravlenie dinamicheskoi sistemoi [Control of a dynamical system]. Moscow, Nauka Publ., 1985, 520 p.
8. Osipov Yu.S., Okhezin S.P. On the theory of differential games in parabolic systems. Sov. Math., Dokl., 1976, vol. 17, pp. 278–282.
9. Okhezin S.P. Differential encounter-evasion game for a parabolic system under integral constraints on the player’s controls. J. Appl. Math. Mech., 1977, vol. 41, no. 2, pp. 194–201. doi: 10.1016/0021-8928(77)90002-8
10. Ukhobotov V.I., Izmest’ev I.V. The problem of controlling the process of heating the rod in the presence of disturbance and uncertainty. IFAC PapersOnLine, 2018, vol. 51, no. 32, pp. 739–742. doi: 10.1016/j.ifacol.2018.11.458
11. Pontryagin L.S. Linear differential games of pursuit. Mathematics of the USSR-Sbornik, 1981, vol. 40, no. 3. pp. 285–303. doi: 10.1070/SM1981v040n03ABEH001815
12. Ukhobotov V.I. Metod odnomernogo proektirovaniya v lineinykh differentsial’nykh igrakh s integral’nymi ogranicheniyami [Method of one-dimensional projecting in linear differential games with integral constraints]. Chelyabinsk: Chelyabinsk State Univ. Publ., 2005, 124 p. ISBN: 5-7271-0725-3 .
13. Ukhobotov V.I. One type differential games with convex goal. Trudy Inst. Mat. i Mekh. UrO RAN, 2010, vol. 16, no. 5, pp. 196–204.
14. Godunov S.K. Uravneniya matematicheskoi fiziki [Equations of mathematical physics]. Moscow: Nauka Publ., 1971, 416 p.
15. Filippov A.F. On Certain questions in the theory of optimal control. J. SIAM Control Ser. A, 1962, vol. 1, no. 1, pp. 76–84. doi: 10.1137/0301006
16. Eidel’man S.D. Parabolic systems. Amsterdam; London: North-Holland Publ. Comp.; Groningen: Wolters–Noordhoff Publ., 1969, 469 p.
17. Mizohata S. The theory of partial differential equations. London: Cambridge Univ. Press, 1973, 490 p. ISBN: 9780521087278 . Translated to Russian under the title Teoriya uravnenii s chastnymi proizvodnymi. Moscow: Mir Publ., 1977, 504 p.
18. Tikhonov A.N., Samarskii A.A. Uravneniya matematicheskoi fiziki [Equations of mathematical physics]. Moscow: Nauka Publ., 1977, 735 p.
19. Il’in A.M. Uravneniya matematicheskoi fiziki [Equations of mathematical physics]. Moscow: Fizmatlit Publ., 2009, 192 p. ISBN: 978-5-9221-1036-5/hbk .
20. Kolmogorov A.N., Fomin S.V. Elements of the theory of functions and functional analysis. (Two volumes in one, translated from the first Russian edition 1957–1961). Martino Fine Books, United States, 2012, 280 p. ISBN: 1614273049 . The 3rd edition of Russian text published in Kolmogorov A.N., Fomin S.V. Elementy teorii funktsii i funktsional’nogo analiza, Moscow: Nauka Publ., 1972, 496 p.
