Control of a parabolic system with disturbances and a convex goal

I. V. Izmestyev ; V. I. Ukhobotov

Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 62 (2023), pp. 30-42.

Résumé

The problem of controlling the heating process of a given number of rods by changing the temperatures at their left ends is considered. Temperatures at the right ends of the rods are formed by disturbances. The density functions of the internal heat sources of the rods are not known exactly, and only the boundaries of the range of their possible values are given. The goal of the choice of a control is to lead the vector of average temperatures of the rods at a fixed time to a convex terminal set. For this problem, necessary and sufficient conditions have been found that must be satisfied by the initial temperatures of the rods so that the goal can be achieved under any admissible realization of disturbances and density functions of internal heat sources. The case of a problem with a possible change in the dynamics of the controlled system is considered.

Détail

Keywords: control, disturbance, parabolic system.

Bibliographie

[1] Osipov Iu.S., “Position control in parabolic systems”, Journal of Applied Mathematics and Mechanics, 41:2 (1977), 187–193 | DOI | MR

[2] Korotkii A.I., Osipov Iu.S., “Approximation in problems of position control of parabolic system”, Journal of Applied Mathematics and Mechanics, 42:4 (1978), 631–637 | DOI | MR

[3] Egorov A.I., Optimal control of thermal and diffusion processes, Nauka, Moscow, 1978

[4] Maksimov V.I., “On the reconstruction of an input disturbance in a reaction–diffusion system”, Computational Mathematics and Mathematical Physics, 63:6 (2023), 990–1000 | DOI | MR | Zbl

[5] Casas E., Yong Jiongmin, “Optimal control of a parabolic equation with memory”, ESAIM: Control, Optimisation and Calculus of Variations, 29 (2023), 23 | DOI | MR | Zbl

[6] Lohéac J., “Nonnegative boundary control of 1D linear heat equations”, Vietnam Journal of Mathematics, 49:3 (2021), 845–870 | DOI | MR | Zbl

[7] Barseghyan V., Solodusha S., “The problem of boundary control of the thermal process in a rod”, Mathematics, 11:13 (2023), 2881 | DOI

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

[9] Zheng Guojie, Li Jun, “Stabilization for the multi-dimensional heat equation with disturbance on the controller”, Automatica, 82 (2017), 319–323 | DOI | MR | Zbl

[10] Feng Hongyinping, Xu Cheng-Zhong, Yao Peng-Fei, “Observers and disturbance rejection control for a heat equation”, IEEE Transactions on Automatic Control, 65:11 (2020), 4957–4964 | DOI | MR

[11] Wang Shanshan, Qi Jie, Diagne Mamadou, “Adaptive boundary control of reaction–diffusion PDEs with unknown input delay”, Automatica, 134 (2021), 109909 | DOI | MR | Zbl

[12] Krasovskii N.N., Control of a dynamical system, Nauka, Moscow, 1985 | MR

[13] Osipov Yu.S., Okhezin S.P., “On the theory of differential games in parabolic systems”, Soviet Mathematics. Doklady, 17 (1976), 278–282 | MR | Zbl | Zbl

[14] Okhezin S.P., “Differential encounter–evasion game for a parabolic system under integral constraints on the player’s controls”, Journal of Applied Mathematics and Mechanics, 41:2 (1977), 194–201 | DOI | MR

[15] Vlasenko L.A., Rutkas A.G., Chikrii A.A., “On a differential game in an abstract parabolic system”, Proceedings of the Steklov Institute of Mathematics, 293:suppl. 1 (2016), 254–269 | DOI | MR

[16] Tukhtasinov M., Ibragimov G., Kuchkarova S., Hasim R.M., “Differential games for an infinite 2-systems of differential equations”, Mathematics, 9:13 (2021), 1467 | DOI

[17] 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

[18] Izmest’ev I.V., Ukhobotov V.I., “On one problem of controlling the heating of a rod system under uncertainty”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp’yuternye Nauki, 32:4 (2022), 546–556 (in Russian) | DOI | MR | Zbl

[19] Ukhobotov V.I., “One type differential games with convex goal”, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 16:5 (2010), 196–204 (in Russian)

[20] Nikol’skij M.S., Pang Chg., “A differential pursuit game with a breakdown in the dynamics”, Differential Equations, 30:11 (1994), 1775–1778 | MR | Zbl | Zbl

[21] Ukhobotov V.I., “On a control problem under a disturbance and possible breakdown”, Proceedings of the Steklov Institute of Mathematics, 307:suppl. 1 (2019), 159–171 | DOI | DOI | MR | Zbl

[22] Godunov S.K., Equations of mathematical physics, Nauka, Moscow, 1971 | MR

[23] Ukhobotov V.I., Izmest’ev I.V., “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, 25:1 (2019), 297–305 (in Russian) | DOI | MR

[24] Pontrjagin L.S., “Linear differential games of pursuit”, Mathematics of the USSR-Sbornik, 40:3 (1981), 285–303 | DOI | MR | Zbl | Zbl

[25] Pshenichnyi B.N., Convex analysis and extremal problems, Nauka, Moscow, 1980 | MR

[26] Petrov N.N., Introduction to convex analysis, Udmurt State University, Izhevsk, 2009