On one boundary control problem for a pseudo-parabolic equation in a two-dimensional domain

Farrukh Dekhkonov; Farrukh Dekhkonov

doi:10.3934/cam.2025001

Communications in Analysis and Mechanics

2025, Volume 17, Issue 1: 1-14. doi: 10.3934/cam.2025001

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Research article

On one boundary control problem for a pseudo-parabolic equation in a two-dimensional domain

Farrukh Dekhkonov ^{1,2
,
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1.
Department of Mathematics, Namangan State University, Namangan 160136, Uzbekistan
2.
Department of Mathematics, National University of Uzbekistan, Tashkent 100174, Uzbekistan

Received: 16 May 2024 Revised: 21 November 2024 Accepted: 07 January 2025 Published: 14 January 2025
35K70, 35K05

In this paper, we consider a boundary control problem associated with a non-homogeneous pseudo-parabolic type equation in a bounded two-dimensional domain. In the part of the bound of the given region, the value of the solution is given, and it is required to find control to get the average value of the solution. The initial-boundary problem is solved by the Fourier method, and the control problem under consideration is analyzed with the Volterra integral equation of the second kind. The control function is found using the Laplace transform method and proved to be admissible.
- pseudo-parabolic equation,
- initial-boundary problem,
- Volterra integral equation,
- admissible control,
- Laplace transform
Citation: Farrukh Dekhkonov. On one boundary control problem for a pseudo-parabolic equation in a two-dimensional domain[J]. Communications in Analysis and Mechanics, 2025, 17(1): 1-14. doi: 10.3934/cam.2025001

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Abstract

In this paper, we consider a boundary control problem associated with a non-homogeneous pseudo-parabolic type equation in a bounded two-dimensional domain. In the part of the bound of the given region, the value of the solution is given, and it is required to find control to get the average value of the solution. The initial-boundary problem is solved by the Fourier method, and the control problem under consideration is analyzed with the Volterra integral equation of the second kind. The control function is found using the Laplace transform method and proved to be admissible.

References

[1]	B. D. Coleman, W. Noll, An Approximation Theorem for Functionals, with Applications in Continuum Mechanics, Arch. Rational Mech. Anal., 6 (1960), 355–370. https://doi.org/10.1007/BF00276168 doi: 10.1007/BF00276168
[2]	S. Albeverio, Sh. A. Alimov, On one time-optimal control problem associated with the heat exchange process, Appl. Math. Opt., 57 (2008), 58–68. https://doi.org/10.1007/s00245-007-9008-7 doi: 10.1007/s00245-007-9008-7
[3]	Sh. A. Alimov, N. M. Komilov, Determining the thermal mode setting parameters based on output data, Diff Equat, 58 (2022), 21–35. https://doi.org/10.1134/S0012266122010049 doi: 10.1134/S0012266122010049
[4]	F. N. Dekhkonov, E. I. Kuchkorov, On the time-optimal control problem associated with the heating process of a thin rod, Lobachevskii. J. Math., 44 (2023), 1134–1144. https://doi.org/10.1134/S1995080223030101 doi: 10.1134/S1995080223030101
[5]	F. N. Dekhkonov, Boundary control problem for the heat transfer equation associated with heating process of a rod, Bulletin of the Karaganda University. Mathematics Series, 110 (2023), 63–71. https://doi.org/10.31489/2023m2/63-71 doi: 10.31489/2023m2/63-71
[6]	H. Xu, Existence and blow-up of solutions for finitely degenerate semilinear parabolic equations with singular potentials, Commun. Anal. Mech., 15 (2023), 132–161. https://doi.org/10.3934/cam.2023008 doi: 10.3934/cam.2023008
[7]	R. Xu, W. Lian, Y. Niu, Global well-posedness of coupled parabolic systems, Sci. China Math., 63 (2020), 321–356. https://doi.org/10.1007/s11425-017-9280-x doi: 10.1007/s11425-017-9280-x
[8]	Yu. V. Egorov, The optimal control in Banach space, Uspekhi Mat. Nauk, 18 (1963), 211–213.
[9]	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
[10]	F. Alabau-Boussouira, P. Cannarsa, G. Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability, J. Evol. Equ, 6 (2006), 161–204. https://doi.org/10.1007/s00028-006-0222-6 doi: 10.1007/s00028-006-0222-6
[11]	B. D. Coleman, R. J. Duffin, V. J. Mizel, Instability, uniqueness, and nonexistence theorems for the equation on a strip, Arch. Rational Mech. Anal., 19 (1965), 100–116. https://doi.org/10.1007/BF00282277 doi: 10.1007/BF00282277
[12]	L. W. White, Point control of pseudoparabolic problems, J. Differ. Equations, 42 (1981), 366–374. https://doi.org/10.1016/0022-0396(81)90110-8 doi: 10.1016/0022-0396(81)90110-8
[13]	S. I. Lyashko, On the solvability of pseudo-parabolic equations, Soviet Math. (Iz. VUZ), 29 (1985), 99–101.
[14]	F. Dekhkonov, On a boundary control problem for a pseudo-parabolic equation, Commun. Anal. Mech., 15 (2023), 289–299. http://doi.org/10.3934/cam.2023015 doi: 10.3934/cam.2023015
[15]	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. http://doi.org/10.12732/ijam.v37i1.9 doi: 10.12732/ijam.v37i1.9
[16]	W. Lian, J. Wang, R. Xu, Global existence and blow up of solutions for pseudo-parabolic equation with singular potential, J. Differ. Equations, 269 (2020), 4914–4959. http://dx.doi.org/10.1016/j.jde.2020.03.047 doi: 10.1016/j.jde.2020.03.047
[17]	R. Xu, J. Su, Global existence and finite-time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264 (2013), 2732–2763. http://dx.doi.org/10.1016/j.jfa.2013.03.010 doi: 10.1016/j.jfa.2013.03.010
[18]	A. N. Tikhonov, A. A. Samarsky, Equations of Mathematical Physics, Nauka, Moscow, 1966.
[19]	M. A. Naimark, Linear differential operators, Nauka, Moscow, 1962.
[20]	V. E. Vladykina, Spectral characteristics of the Sturm-Liouville operator under minimal restrictions on smoothness of coefficients, Moscow Univ. Math. Bull., 74 (2019), 235–240. https://doi.org/10.3103/S0027132219060044 doi: 10.3103/S0027132219060044

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