Geodesic
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 mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 25 (2019) no. 1, pp. 297-305 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

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. 
@article{TIMM_2019_25_1_a21,
     author = {V. I. Ukhobotov and I. V. Izmestyev},
     title = {A control problem for a rod heating process with unknown temperature at the right end and unknown density of the heat source},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {297--305},
     year = {2019},
     volume = {25},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2019_25_1_a21/}
}
TY  - JOUR
AU  - V. I. Ukhobotov
AU  - I. V. Izmestyev
TI  - A control problem for a rod heating process with unknown temperature at the right end and unknown density of the heat source
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2019
SP  - 297
EP  - 305
VL  - 25
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TIMM_2019_25_1_a21/
LA  - ru
ID  - TIMM_2019_25_1_a21
ER  - 
%0 Journal Article
%A V. I. Ukhobotov
%A I. V. Izmestyev
%T A control problem for a rod heating process with unknown temperature at the right end and unknown density of the heat source
%J Trudy Instituta matematiki i mehaniki
%D 2019
%P 297-305
%V 25
%N 1
%U http://geodesic.mathdoc.fr/item/TIMM_2019_25_1_a21/
%G ru
%F TIMM_2019_25_1_a21
V. I. Ukhobotov; I. V. Izmestyev. 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 mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 25 (2019) no. 1, pp. 297-305. http://geodesic.mathdoc.fr/item/TIMM_2019_25_1_a21/

[1] Osipov Yu.S., “Pozitsionnoe upravlenie v parabolicheskikh sistemakh”, Prikl. matematika i mekhanika, 41:2 (1977), 195–201 | MR

[2] Korotkii A.I., Osipov Yu.S., “Approksimatsiya v zadachakh pozitsionnogo upravleniya parabolicheskimi sistemami”, Prikl. matematika i mekhanika, 42:4 (1978), 599–605 | MR

[3] Egorov A.I., Optimalnoe upravlenie teplovymi i diffuzionnymi protsessami, Nauka, M., 1978, 464 pp.

[4] Vasilev F.P., Metody resheniya ekstremalnykh zadach, Nauka, M., 1981, 400 pp.

[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., 428 (2015), 1193–1201 | DOI | MR | Zbl

[6] Dai J., Ren B., “UDE-based robust boundary control of heat equation with unknown input disturbance”, IFAC PapersOnLine, 50:1 (2017), 11403–11408 | DOI

[7] Krasovskii N.N., Upravlenie dinamicheskoi sistemoi, Nauka, M., 1985, 520 pp.

[8] Osipov Yu.S., Okhezin S.P., “K teorii differentsialnykh igr v parabolicheskikh sistemakh”, Dokl. AN, 226:6 (1976), 1267–1270 | MR | Zbl

[9] Okhezin S.P., “Differentsialnaya igra sblizheniya-ukloneniya dlya parabolicheskoi sistemy s integralnymi ogranicheniyami na upravleniya igrokov”, Prikl. matematika i mekhanika, 41:2 (1977), 202–209 | MR

[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, 51:32 (2018), 739–742 | DOI

[11] Pontryagin L.S., “Lineinye differentsialnye igry presledovaniya”, Mat. sb. Novaya seriya, 112:3 (1980), 307–330 | MR | Zbl

[12] Ukhobotov V.I., Metod odnomernogo proektirovaniya v lineinykh differentsialnykh igrakh s integralnymi ogranicheniyami, ucheb. posobie, Izd-vo Chelyab. gos. un-ta, Chelyabinsk, 2005, 124 pp.

[13] Ukhobotov V.I., “Odnotipnye differentsialnye igry s vypukloi tselyu”, Tr. In-ta matematiki i mekhaniki Uro RAN, 16:5 (2010), 196–204

[14] Godunov S.K., Uravneniya matematicheskoi fiziki, Nauka, M., 1971, 416 pp.

[15] Filippov A.F., “O nekotorykh voprosakh teorii optimalnogo regulirovaniya”, Vestn. MGU. Ser. Matematika, mekhanika, 1959, no. 2, 25–32 | Zbl

[16] Eidelman S.D., Parabolicheskie sistemy, Nauka, M., 1964, 444 pp. | MR

[17] Mizokhata S., Teoriya uravnenii s chastnymi proizvodnymi, Mir, M., 1977, 504 pp.

[18] Tikhonov A.N., Samarskii A.A., Uravneniya matematicheskoi fiziki, Nauka, M., 1977, 735 pp. | MR

[19] Ilin A.M., Uravneniya matematicheskoi fiziki, ucheb. posobie, Fizmatlit, M., 2009, 192 pp.

[20] Kolmogorov A.N., Fomin S.V., Elementy teorii funktsii i funktsionalnogo analiza, Nauka, M., 1972, 496 pp.