mathematicsВестник российских университетов. МатематикаRussian Universities Reports. Mathematics2686-96672782-3342Тамбовский государственный университет имени Г.Р. Державина10.20310/2686-9667-2025-30-152-322-337mathematics-108Research ArticleНАУЧНЫЕ СТАТЬИORIGINAL ARTICLESО задаче управления для псевдопараболического уравнения с инволюцией в ограниченной областиOn the control problem for a pseudo-parabolic equation with involution in a bounded domainhttps://orcid.org/0000-0003-4747-8557ДехконовФаррух НуриддинDekhkonovFarrukh N.f.n.dehqonov@mail.ruhttps://orcid.org/0000-0001-7735-6484ТурметовБатирхан ХудайбергеновичTurmetovBatirkhan Kh.batirkhan.turmetov@ayu.edu.kzНаманганский государственный университетРоссияNamangan State UniversityRussian FederationМеждународный казахско-турецкий университет имени Ходжи Ахмеда Ясави; Университет АльфраганусРоссияKhoja Akhmet Yassawi International Kazakh-Turkish University; Alfraganus UniversityRussian Federation20251212202530152322337Copyright © Дехконов Ф.Н., Турметов Б.Х., 20252025Дехконов Ф.Н., Турметов Б.Х.Dekhkonov F.N., Turmetov B.K.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://mathematics.elpub.ru/jour/article/view/108

В данной работе рассматривается задача управления для псевдопараболического уравнения с оператором инволюции в ограниченной области. Получено обобщенное решение соответствующей начально-краевой задачи. Путем введения дополнительного интегрального условия задача управления сведена к интегральному уравнению Вольтерра первого рода. Для того чтобы показать, что интегральное уравнение имеет решение, получены некоторые оценки для ядра этого интегрального уравнения. С помощью метода преобразования Лапласа показано существование решения интегрального уравнения и доказана допустимость функции управления.

This paper considers a control problem for a pseudo-parabolic equation with an involution operator in a bounded domain. A generalized solution to the corresponding initial boundary value problem is obtained. By introducing an additional integral condition, the control problem is reduced to a Volterra integral equation of the first kind. To show that the integral equation has a solution, some estimates are obtained for the kernel of this integral equation. The existence of a solution to the integral equation is shown using the Laplace transform method and the admissibility of the control function is proved. Keywords: pseudo-parabolic equation, Volterra integral equation, admissible control, initial boundary value problem, Laplace transform, weight function

псевдопараболическое уравнениеинтегральное уравнение Вольтеррадопустимое управлениеначально-краевая задачапреобразование Лапласавесовая функцияpseudo-parabolic equationVolterra integral equationadmissible controlinitial boundary value problemLaplace transformweight functionИсследование выполнено при финансовой поддержке Комитета науки Министерства науки и высшего образования Республики Казахстан (грант № AP23488086).The research was supported by the Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan (grant no. AP23488086).ReferencesB.D. Coleman, W. Noll, “An approximation theorem for functionals, with applications in continuum mechanics”, Arch. Rational Mech. Anal., 6 (1960), 355–370.B.D. Coleman, W. Noll, “An approximation theorem for functionals, with applications in continuum mechanics”, Arch. Rational Mech. Anal., 6 (1960), 355–370.P. Chen, M. Gurtin, “On a theory of heat conduction involving two temperatures”, Z. Angew. Math. Phys., 19 (1968), 614–627.P. Chen, M. Gurtin, “On a theory of heat conduction involving two temperatures”, Z. Angew. Math. Phys., 19 (1968), 614–627.L.W. White, “Controllability properties of pseudo-parabolic boundary control problems”, SIAM J. Control and Optimization, 18 (1980), 534–539.L.W. White, “Controllability properties of pseudo-parabolic boundary control problems”, SIAM J. Control and Optimization, 18 (1980), 534–539.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.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.L.W. White, “Point control of pseudoparabolic problems”, Journal of Differential Equations, 42:3 (1981), 366–374.L.W. White, “Point control of pseudoparabolic problems”, Journal of Differential Equations, 42:3 (1981), 366–374.A. Friedman, “Optimal control for parabolic equations”, J. Math. Anal. Appl., 18:3 (1967), 479–491.A. Friedman, “Optimal control for parabolic equations”, J. Math. Anal. Appl., 18:3 (1967), 479–491.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.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.Yu.V. Egorov, “The optimal control in Banach space”, Uspekhi Mat. Nauk., 18 (1963), 211–213.Yu.V. Egorov, “The optimal control in Banach space”, Uspekhi Mat. Nauk., 18 (1963), 211–213.A.V. Fursikov, Optimal Control of Distributed Systems. Theory and Applications: Translations of Mathematical Monographs, 187, Amer. Math. Soc., Providence, Rhode Island, 2000, 305 pp.A.V. Fursikov, Optimal Control of Distributed Systems. Theory and Applications: Translations of Mathematical Monographs, 187, Amer. Math. Soc., Providence, Rhode Island, 2000, 305 pp.J.L. Lions, Contrґole optimal de syst`emes gouvernґes par des ґequations aux dґerivґees partielles, Dunod Gauthier–Villars, Paris, 1968.J.L. Lions, Contrґole optimal de syst`emes gouvernґes par des ґequations aux dґerivґees partielles, Dunod Gauthier–Villars, Paris, 1968.G. Schmidt, “The “Bang–Bang” principle for the time-optimal problem in boundary control of the heat equation”, SIAM Journal on Control and Optimization, 18 (1980), 101–107.G. Schmidt, “The “Bang–Bang” principle for the time-optimal problem in boundary control of the heat equation”, SIAM Journal on Control and Optimization, 18 (1980), 101–107.S. Albeverio, Sh.A. Alimov, “On one time-optimal control problem associated with the heat exchange process”, Appl. Math. Opt., 57 (2008), 58–68.S. Albeverio, Sh.A. Alimov, “On one time-optimal control problem associated with the heat exchange process”, Appl. Math. Opt., 57 (2008), 58–68.Z.K. Fayazova, “Boundary control of the heat transfer process in the space”, Russian Mathematics, 12 (2019), 82–90.Z.K. Fayazova, “Boundary control of the heat transfer process in the space”, Russian Mathematics, 12 (2019), 82–90.F.N. Dekhkonov, “On the time-optimal control problem for a heat equation”, Bulletin of the Karaganda University Mathematics Series, 111 (2023), 28–38.F.N. Dekhkonov, “On the time-optimal control problem for a heat equation”, Bulletin of the Karaganda University Mathematics Series, 111 (2023), 28–38.B. Allal, G. Fragnelli, J. Salhi, “Null controllability for degenerate parabolic equations with a nonlocal space term”, Discrete and Continuous Dynamical Systems - S, 17 (2024), 1821–1856.B. Allal, G. Fragnelli, J. Salhi, “Null controllability for degenerate parabolic equations with a nonlocal space term”, Discrete and Continuous Dynamical Systems - S, 17 (2024), 1821–1856.F. Dekhkonov, “On one boundary control problem for a pseudo-parabolic equation in a twodimensional domain”, Communications in Analysis and Mechanics, 17 (2025), 1–14.F. Dekhkonov, “On one boundary control problem for a pseudo-parabolic equation in a twodimensional domain”, Communications in Analysis and Mechanics, 17 (2025), 1–14.F.N. Dekhkonov, “On the control problem associated with a pseudo-parabolic type equation in an one-dimensional domain”, International Journal of Applied Mathematics, 37:1 (2024), 109–118.F.N. Dekhkonov, “On the control problem associated with a pseudo-parabolic type equation in an one-dimensional domain”, International Journal of Applied Mathematics, 37:1 (2024), 109–118.B.K. Turmetov, B.J. Kadirkulov, “An inverse problem for a parabolic equation with involution”, Lobachevskii J. of Math., 42 (2021), 3006–3015.B.K. Turmetov, B.J. Kadirkulov, “An inverse problem for a parabolic equation with involution”, Lobachevskii J. of Math., 42 (2021), 3006–3015.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.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.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., 1 (2022), 99.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., 1 (2022), 99.B. Ahmad, A. Alsaedi, M. Kirane, R. Tapdigoglu, “An inverse problem for space and time fractional evolution equations with an involution perturbation”, Quaest. Math., 40 (2017), 151–160.B. Ahmad, A. Alsaedi, M. Kirane, R. Tapdigoglu, “An inverse problem for space and time fractional evolution equations with an involution perturbation”, Quaest. Math., 40 (2017), 151–160.A. Kopzhassarova, A. Sarsenbi, “Basis properties of eigenfunctions of second-order differential operators with involution”, Abstr. Appl. Anal., 2012, 576843.A. Kopzhassarova, A. Sarsenbi, “Basis properties of eigenfunctions of second-order differential operators with involution”, Abstr. Appl. Anal., 2012, 576843.M. Kirane, A.A. Sarsenbi, “Solvability of mixed problems for a fourth-order equation with involution and fractional derivative”, Fractal Fract., 7 (2023), 131.M. Kirane, A.A. Sarsenbi, “Solvability of mixed problems for a fourth-order equation with involution and fractional derivative”, Fractal Fract., 7 (2023), 131.B. Turmetov, V. Karachik, “On solvability of some inverse problems for a nonlocal fourth-order parabolic equation with multiple involution”, AIMS Mathematics, 9 (2024), 6832–6849.B. Turmetov, V. Karachik, “On solvability of some inverse problems for a nonlocal fourth-order parabolic equation with multiple involution”, AIMS Mathematics, 9 (2024), 6832–6849.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 and Fractional, 7:5 (2023), 404.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 and Fractional, 7:5 (2023), 404.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 (2019), 669–681.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 (2019), 669–681.F.N. Dekhkonov, “On the control problem for a heat conduction equation with involution in a two-dimensional domain”, Lobachevskii J. Math., 46:2 (2025), 613–623.F.N. Dekhkonov, “On the control problem for a heat conduction equation with involution in a two-dimensional domain”, Lobachevskii J. Math., 46:2 (2025), 613–623.F.N. Dekhkonov, “Boundary control problem for a parabolic equation with involution”, Eurasian Journal of Mathematical and Computer Applications, 12 (2024), 22–34.F.N. Dekhkonov, “Boundary control problem for a parabolic equation with involution”, Eurasian Journal of Mathematical and Computer Applications, 12 (2024), 22–34.S.G. Mikhlin, Linear Partial Differential Equations, Vysshaya Shkola Publ., Moscow, 1977.S.G. Mikhlin, Linear Partial Differential Equations, Vysshaya Shkola Publ., Moscow, 1977.B. Turmetov, V. Karachik, “On eigenfunctions and eigenvalues of a nonlocal Laplace operator with involution in a parallelepiped”, AIP Conf. Proc., Sixth International Conference of Mathematical Sciences (ICMS 2022) (Istanbul, Turkey, 20–24 July 2022), 2879, 2023.B. Turmetov, V. Karachik, “On eigenfunctions and eigenvalues of a nonlocal Laplace operator with involution in a parallelepiped”, AIP Conf. Proc., Sixth International Conference of Mathematical Sciences (ICMS 2022) (Istanbul, Turkey, 20–24 July 2022), 2879, 2023.O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Uraltseva, Linear and Quasi-Linear Equations of Parabolic Type, Nauka Publ., Moscow, 1967.O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Uraltseva, Linear and Quasi-Linear Equations of Parabolic Type, Nauka Publ., Moscow, 1967.Sh.A. Alimov, F.N. Dekhkonov, “On a control problem associated with fast heating of a thin rod”, Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences, 2 (2019), 1–14.Sh.A. Alimov, F.N. Dekhkonov, “On a control problem associated with fast heating of a thin rod”, Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences, 2 (2019), 1–14.F. Dekhkonov, W. Li, “On the boundary control problem associated with a fourth order parabolic equation in a two-dimensional domain”, Discrete and Continuous Dynamical Systems - S, 17 (2024), 2478–2488.F. Dekhkonov, W. Li, “On the boundary control problem associated with a fourth order parabolic equation in a two-dimensional domain”, Discrete and Continuous Dynamical Systems - S, 17 (2024), 2478–2488.

The authors declare that there are no conflicts of interest present.