On the time-optimal control problem associated with fourth-order parabolic equation in a ball
DOI:
https://doi.org/10.70474/3pkr9v84Keywords:
fourth order parabolic equation, Volterra integral equation, admissible control, thin film, minimal timeAbstract
In this paper, we consider the optimal time control problem for a fourth-order parabolic equation describing the growth process of a thin film. The control function is given on a certain part of the boundary. The solution of the control problem is reduced to a first-order Volterra integral equation. As a result, it is proved that there is a control function necessary to achieve a given average thickness of the thin film, and an optimal estimate for the minimum time is found.
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Albeverio S., Alimov Sh.A. textit{On one time-optimal control problem associated with the heat exchange process}, Appl. Math. Optim. 57:1 (2008), 58–68.
Alimov Sh.A., Dekhkonov F.N. textit{On the time-optimal control of the heat exchange process}, Uzbek Math. J. No. 2 (2019), 4–17.
Alimov Sh.A., Ibragimov G. textit{Time optimal control problem with integral constraint for the heat transfer process}, Eurasian Math. J. 15:1 (2024), 8–22.
Chen N., Wang Y., Yang D. textit{Time-varying bang-bang property of time optimal controls for heat equation and its applications}, Systems Control Lett. 112 (2018), 18–23. https://doi.org/10.1016/j.sysconle.2017.12.008
Chen Y. textit{Global dynamical behavior of solutions for finite degenerate fourth-order parabolic equations with mean curvature nonlinearity}, Commun. Anal. Mech. 15:4 (2023), 658–694.
Carreno N., Guzman P. textit{On the cost of null controllability of a fourth-order parabolic equation}, J. Differ. Equ. 261:11 (2016), 6485–6520.
Dekhkonov F.N. textit{On the time-optimal control problem associated with the epitaxial growth of thin film}, Uzbek Math. J. No. 2 (2019), 37–47.
Dekhkonov F.N. textit{On the control problem associated with the heating process}, Math. Notes of NEFU 29:4 (2022), 62–71.
Dekhkonov F.N., Kuchkorov E.I. textit{On the time-optimal control problem associated with the heating process of a thin rod}, Lobachevskii J. Math. 44:3 (2023), 1134–1144. https://doi.org/10.1134/S1995080223030101
Dekhkonov F.N. textit{Boundary control problem for the heat transfer equation associated with heating process of a rod}, Bull. Karaganda Univ. Math. Ser. 110:2 (2023), 63–71. https://doi.org/10.31489/2023M2/63-71
Dekhkonov F.N. textit{On the time-optimal control problem for a heat equation}, Bull. Karaganda Univ. Math. Ser. 111:3 (2023), 28–38. https://doi.org/10.31489/2023m3/28-38
Dekhkonov F.N., Li W. textit{On the boundary control problem associated with a fourth order parabolic equation in a two-dimensional domain}, Discrete Contin. Dyn. Syst. S 17:8 (2024), 2478–2488.
Dekhkonov F.N. textit{On the time-optimal control problem for a fourth order parabolic equation in a two-dimensional domain}, Bull. Karaganda Univ. Math. Ser. 114:1 (2024), 57–70.
Duffin R.J. textit{On a question of Hadamard concerning super-biharmonic functions}, J. Math. Phys. 27:1 (1949), 253–258.
Egorov Yu.V. textit{Optimal control in Banach spaces}, Dokl. Akad. Nauk SSSR 150:2 (1963), 241–244. (in Russian)
Friedman A. textit{Differential equations of parabolic type}, Englewood Cliffs, NJ (1964). https://doi.org/10.1016/0022-247X(67)90040-6
Fattorini H.O., Russell D.L. textit{Exact controllability theorems for linear parabolic equations in one space dimension}, Arch. Ration. Mech. Anal. 43:4 (1971), 272–292.
Fattorini H.O. textit{Time-optimal control of solutions of operational differential equations}, SIAM J. Control 2:1 (1964), 49–65.
Fayazova Z.K. textit{Boundary control of the heat transfer process in the space}, Russian Math. 63:12 (2019), 71–79. https://doi.org/10.26907/0021-3446-2019-12-82-90
Fursikov A.V. textit{Optimal Control of Distributed Systems: Theory and Applications}, Amer. Math. Soc., Providence, RI, Transl. Math. Monogr. 187 (2000).
Guo Y.-J.L. textit{Null boundary controllability for a fourth order parabolic equation}, Taiwan. J. Math. 6:2 (2002), 421–431.
Grunau H.C., Sweers G. textit{Sign change for the Green function and the first eigenfunction of equations of clamped-plate type}, Arch. Ration. Mech. Anal. 150:2 (1999), 179–190.
Garabedian P.R. textit{A partial differential equation arising in conformal mapping}, Pacific J. Math. 1:3 (1951), 485–524.
Hadamard J. textit{Sur certains cas interessants du probleme biharmonique}, OEuvres de Jacques Hadamard, Tome III, CNRS, Paris (1968), 1297–1299. (Reprint of: Atti IV Congr. Intern. Mat., Rome (1908), 12–14.)
Jentzsch P. textit{Uber Integralgleichungen mit positivem Kern}, J. Reine Angew. Math. 141 (1912), 235–244.
King B.B., Stein O., Winkler M. textit{A fourth-order parabolic equation modeling epitaxial thin film growth}, J. Math. Anal. Appl. 286:2 (2003), 459–490.
Krein M.G., Rutman M.A. textit{Linear operators leaving invariant a cone in a Banach space}, Uspekhi Mat. Nauk 3:1 (1948), 3–95. (English transl.: Amer. Math. Soc. Transl., 1950)
Lions J.L. textit{Controle optimal de systemes gouvernes par des equations aux derivees partielles}, Dunod Gauthier-Villars, Paris (1968).
Laroche B., Martin P., Rouchon P. textit{Motion planning for the heat equation}, Int. J. Robust Nonlinear Control 10:8 (2000), 629–643.
Tikhonov A.N., Samarskii A.A. textit{Equations of Mathematical Physics}, Nauka, Moscow (1966).
Wang G., Wang L. textit{The bang-bang principle of time optimal controls for the heat equation with internal controls}, Systems Control Lett. 56:9 (2007), 709–713.
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