On one time-optimal control problem for a parabolic equation with involution in a bounded domain

Farrukh Dekhkonov; Batirkhan Turmetov; Farrukh Dekhkonov; Batirkhan Turmetov

doi:10.3934/math.2025916

AIMS Mathematics

2025, Volume 10, Issue 9: 20531-20549. doi: 10.3934/math.2025916

Previous Article Next Article

Research article Special Issues

On one time-optimal control problem for a parabolic equation with involution in a bounded domain

Farrukh Dekhkonov ¹,
Batirkhan Turmetov ^{2,3
,
,}

1.
Department of Mathematics, Namangan State University, Namangan, Uzbekistan
2.
Department of Mathematics, Khoja Akhmet Yassawi International Kazakh-Turkish University, Turkistan, Kazakhstan
3.
Department of Mathematics, Alfraganus University, Tashkent, Uzbekistan

Received: 12 June 2025 Revised: 17 August 2025 Accepted: 29 August 2025 Published: 08 September 2025
MSC : 35K05, 35K15

In this paper, we study the problem of time-optimal control for the parabolic equation with involution in a multidimensional parallelepiped domain. A generalized solution to the initial boundary value problem is found, and the control problem is reduced to a first-order Volterra integral equation. To prove the existence and uniqueness of the solution to this integral equation, necessary estimates are obtained for its kernel. The existence of a solution to the integral equation, i.e., the admissibility of the control function, is proven, and an optimal estimate of the minimum time required to heat the domain to a certain average temperature is found.
- parabolic equation,
- Volterra integral equation,
- admissible control,
- minimal time,
- eigenfunction,
- involution
Citation: Farrukh Dekhkonov, Batirkhan Turmetov. On one time-optimal control problem for a parabolic equation with involution in a bounded domain[J]. AIMS Mathematics, 2025, 10(9): 20531-20549. doi: 10.3934/math.2025916

Related Papers:

Abstract

In this paper, we study the problem of time-optimal control for the parabolic equation with involution in a multidimensional parallelepiped domain. A generalized solution to the initial boundary value problem is found, and the control problem is reduced to a first-order Volterra integral equation. To prove the existence and uniqueness of the solution to this integral equation, necessary estimates are obtained for its kernel. The existence of a solution to the integral equation, i.e., the admissibility of the control function, is proven, and an optimal estimate of the minimum time required to heat the domain to a certain average temperature is found.

References

[1]	A. Friedman, Optimal control for parabolic equations, J. Math. Anal. Appl., 18 (1967), 479–491. https://doi.org/10.1016/0022-247X(67)90040-6 doi: 10.1016/0022-247X(67)90040-6
[2]	H. O. Fattorini, D. L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Rational Mech. Anal., 43 (1971), 272–292. https://doi.org/10.1007/BF00250466 doi: 10.1007/BF00250466
[3]	H. O. Fattorini, Time-optimal control of solutions of operational differential equations, J. SIAM Control, 2 (1964), 49–65. https://doi.org/10.1137/0302005 doi: 10.1137/0302005
[4]	Yu. V. Egorov, The optimal control in Banach space (in Russian), Uspekhi Mat. Nauk, 18 (1963), 211–213.
[5]	N. Chen, Y. Wang, D. H. Yang, Time-varying bang-bang property of time optimal controls for heat equation and its applications, Syst. Control Lett., 112 (2018), 18–23. https://doi.org/10.1016/j.sysconle.2017.12.008 doi: 10.1016/j.sysconle.2017.12.008
[6]	S. Albeverio, S. A. Alimov, On a time-optimal control problem associated with the heat exchange process, Appl. Math. Optim., 57 (2008), 58–68. https://doi.org/10.1007/s00245-007-9008-7 doi: 10.1007/s00245-007-9008-7
[7]	S. Alimov, On the null-controllability of the heat exchange process, Eurasian Math. J., 2 (2011), 5–19.
[8]	S. Alimov, G. Ibragimov, Time optimal control problem with integral constraint for the heat transfer process, Eurasian Math. J., 15 (2024), 8–22.
[9]	Z. K. Fayazova, Boundary control of the heat transfer process in the space, Russian Math., 63 (2019), 71–79. https://doi.org/10.3103/S1066369X19120089 doi: 10.3103/S1066369X19120089
[10]	F. N. Dekhkonov, Boundary control associated with a parabolic equation, J. Math. Comput. Sci., 33 (2024), 146–154. https://doi.org/10.22436/jmcs.033.02.03 doi: 10.22436/jmcs.033.02.03
[11]	F. Dekhkonov, W. Li, On the boundary control problem associated with a fourth order parabolic equation in a two-dimensional domain, Discrete Cont. Dyn. Syst.- S, 17 (2024), 2478–2488. https://doi.org/10.3934/dcdss.2024004 doi: 10.3934/dcdss.2024004
[12]	N. Arada, J. P. Raymond, Time optimal problems with Dirichlet boundary conditions, Discrete Cont. Dyn. Syst., 9 (2003), 1549–1570. https://doi.org/10.3934/dcds.2003.9.1549 doi: 10.3934/dcds.2003.9.1549
[13]	G. Wang, The existence of time optimal control of semilinear parabolic equations, Syst. Control Lett., 53 (2004), 171–175. https://doi.org/10.1016/j.sysconle.2004.04.002 doi: 10.1016/j.sysconle.2004.04.002
[14]	F. N. Dekhkonov, On the control problem associated with a pseudo-parabolic type equation in an one-dimensional domain, Int. J. Appl. Math., 37 (2024), 109–118. https://doi.org/10.12732/ijam.v37i1.9 doi: 10.12732/ijam.v37i1.9
[15]	F. Dekhkonov, On one boundary control problem for a pseudo-parabolic equation in a two-dimensional domain, Commun. Anal. Mech., 17 (2025), 1–14. https://doi.org/10.3934/cam.2025001 doi: 10.3934/cam.2025001
[16]	B. K. Turmetov, B. J. Kadirkulov, An inverse problem for a parabolic equation with involution, Lobachevskii J. Math., 42 (2021), 3006–3015. https://doi.org/10.1134/S1995080221120350 doi: 10.1134/S1995080221120350
[17]	B. K. Turmetov, B. J. Kadirkulov, On the solvability of an initial-boundary value problem for a fractional heat equation with involution, Lobachevskii J. Math., 43 (2022), 249–262. https://doi.org/10.1134/S1995080222040217 doi: 10.1134/S1995080222040217
[18]	E. Mussirepova, A. Sarsenbi, A. Sarsenbi, The inverse problem for the heat equation with reflection of the argument and with a complex coefficient, Bound. Value Probl., 2022 (2022), 99. https://doi.org/10.1186/s13661-022-01675-1 doi: 10.1186/s13661-022-01675-1
[19]	B. Ahmad, A. Alsaedi, M. Kirane, R. G. Tapdigoglu, An inverse problem for space and time fractional evolution equations with an involution perturbation, Quaest. Math., 40 (2017), 151–160. https://doi.org/10.2989/16073606.2017.1283370 doi: 10.2989/16073606.2017.1283370
[20]	A. Kopzhassarova, A. Sarsenbi, Basis properties of eigenfunctions of second-order differential operators with involution, Abstr. Appl. Anal., 2012 (2012), 1–6. https://doi.org/10.1155/2012/576843 doi: 10.1155/2012/576843
[21]	M. Kirane, A. A. Sarsenbi, Solvability of mixed problems for a fourth-order equation with involution and fractional derivative, Fractal Fract., 7 (2023), 1–12. https://doi.org/10.3390/fractalfract7020131 doi: 10.3390/fractalfract7020131
[22]	B. Turmetov, V. Karachik, On solvability of some inverse problems for a nonlocal fourth-order parabolic equation with multiple involution, AIMS Math., 9 (2024), 6832–6849. https://doi.org/10.3934/math.2024333 doi: 10.3934/math.2024333
[23]	M. Muratbekova, B. Kadirkulov, M. Koshanova, B. Turmetov, On solvability of some inverse problems for a fractional parabolic equation with a nonlocal biharmonic operator, Fractal Fract., 7 (2023), 404. https://doi.org/10.3390/fractalfract7050404 doi: 10.3390/fractalfract7050404
[24]	N. Al-Salti, M. Kirane, B. T. Torebek, On a class of inverse problems for a heat equation with involution perturbation, Hacet. J. Math. Stat., 48 (2016), 669–681.
[25]	F. N. Dekhkonov, On the control problem for a heat conduction equation with involution in a two-dimensional domain, Lobachevskii J. Math., 46 (2025), 613–623. https://doi.org/10.1134/S199508022560013X doi: 10.1134/S199508022560013X
[26]	F. N. Dekhkonov, Boundary control problem for a parabolic equation with involution, Eurasian J. Math. Comput. Appl., 12 (2024), 22–34. https://doi.org/10.32523/2306-6172-2024-12-3-22-34 doi: 10.32523/2306-6172-2024-12-3-22-34
[27]	S. G. Mikhlin, Linear partial differential equations (in Russian), Vysshaya Shkola, Moskva, 1977.
[28]	B. Turmetov, V. Karachik, On eigenfunctions and eigenvalues of a nonlocal Laplace operator with involution in a parallelepiped, AIP Conf. Proc., 2879 (2023), 040005. https://doi.org/10.1063/5.0175246 doi: 10.1063/5.0175246
[29]	O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Uraltseva, Linear and quasi-linear equations of parabolic type (in Russian), Nauka, Moscow, 1967.
[30]	S. A. Alimov, F. N. Dekhkonov, On a control problem associated with fast heating of a thin rod, Bull. Natl Univ. Uzb.: Math. Nat. Sci., 2 (2019), 1–14.

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)