kaz29Вестник Казахстанско-Британского технического университетаHerald of the Kazakh-British Technical University1998-66882959-8109Казахстанско-Британский Технический Университет10.55452/1998-6688-2026-23-3-55-69kaz29-3175Research ArticleМАТЕМАТИЧЕСКИЕ НАУКИMATHEMATICAL SCIENCESО ЗАДАЧЕ УПРАВЛЕНИЯ ДЛЯ ОДНОМЕРНОГО ПСЕВДОПАРАБОЛИЧЕСКОГО УРАВНЕНИЯ С НЕЛОКАЛЬНЫМ ГРАНИЧНЫМ УСЛОВИЕМON THE CONTROL PROBLEM FOR A ONE-DIMENSIONAL PSEUDO-PARABOLIC EQUATION WITH A NONLOCAL BOUNDARY CONDITIONhttps://orcid.org/0000-0003-4747-8557DekhkonovF. N.DekhkonovF. N.

Доктор физико-математических наук, доцент, кафедра математики

Наманган

 

Doctor of Physical and Mathematical Sciences, Associate Professor

Namangan

f.n.dehqonov@mail.ruUsubjonovА. А.UsubjonovA. A.

Докторант PhD, кафедра математики

Наманган

 

PhD Student

Namangan

adhamusubjonov@gmail.comНаманганский государственный университетУзбекистанDepartment of Mathematics, Namangan State UniversityUzbekistan2026240920262335569Copyright © Dekhkonov F.N., Usubjonov А.А., 20262026Dekhkonov F.N., Usubjonov А.А.Dekhkonov F.N., Usubjonov A.A.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://vestnik.kbtu.edu.kz/jour/article/view/3175

В настоящей статье исследуется задача управления для одномерного псевдопараболического уравнения с нелокальным граничным условием типа Бицадзе-Самарского. Управляющая функция входит в правую часть уравнения в виде распределенного источника, и цель состоит в том, чтобы перевести систему из нулевого начального состояния так, чтобы средневзвешенная температура следовала заданной траектории. Уравнение содержит член третьего порядка со смешанной производной, который учитывает эффекты памяти или вязкоупругости в теплопроводности, что особенно важно для материалов с внутренней структурой, таких как полимеры, биологические ткани и гранулированные среды. С помощью спектрального разложения и биортогонального базиса задача сводится к интегральному уравнению Вольтерра первого рода. Доказано, что ядро этого интегрального уравнения положительно и непрерывно. При соответствующих предположениях о весовой функции и пространственном распределении управления доказано существование допустимого управления для класса желаемых траекторий с достаточно малой нормой. Для решения интегрального уравнения и получения оценок управляющей функции используется метод преобразования Лапласа. Результаты количественно характеризуют влияние нелокального граничного условия и параметра памяти на управляемость системы.

This paper studies a control problem for a one-dimensional pseudo-parabolic equation with a nonlocal boundary condition of the Bitsadze-Samarskii type. The control function appears as a distributed source term on the right-hand side of the equation, and the ob jective is to steer the system from the zero initial state so that the weighted average temperature follows a prescribed trajectory. The equation contains a third-order mixed derivative term that accounts for memory or viscoelastic effects i n h eat conduction, which is particularly relevant for materials with internal structure such as polymers, biological tissues, and granular media. Using spectral decomposition and a biorthogonal basis, the problem is reduced to a Volterra integral equation of the first kind. The kernel of this integral equation is proved to be p ositive and continuous. Under suitable assumptions on the weight function and the spatial distribution of the control, the existence of an admissible control is established for a class of desired trajectories with sufficiently small norm. The Laplace transform method is employed to solve the integral equation and to obtain bounds on the control function. The results quantify the influence o f t he n onlocal b oundary c ondition a nd the memory parameter on the controllability of the system.

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