MSC: 93C10, 93B03, 93B35
DOI: 10.21538/0134-4889-2022-28-3-274-284
Полный текст статьи (Full text)
The control systems described by nonlinear differential equations and integral constraints on the control functions are studied. Admissible control functions are chosen from a closed ball of the space $L_p,$ $p\in (1,\infty]$, with radius $r$ and centered at the origin. It is proved that the set of trajectories of the system is continuous at $p=\infty$ with respect to the Hausdorff pseudometric. It is shown that every trajectory is robust with respect to the fast and full consumption of the remaining control resource which implies that to achieve the desired result, it is advisable to spend the available control resource in small portions. This allows to prove that every trajectory can be approximated by the trajectory, generated by full consumption of the control resource.
Keywords: nonlinear control system, set of trajectories, integral constraint, geometric constraint, Hausdorff continuity, robustness
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Received May 10, 2022
Revised June 9, 2022
Accepted June 13, 2022
Nesir Huseyin, Associate Professor, Sivas Cumhuriyet University, Faculty of Education, 58140 Sivas, Turkey, e-mail: nhuseyin@cumhuriyet.edu.tr
Anar Huseyin, Associate Professor, Sivas Cumhuriyet University, Faculty of Science, 58140 Sivas, Turkey, e-mail: ahuseyin@cumhuriyet.edu.tr
Khalik Garakishi oglu Guseinov, Dr. Phys.-Math. Sci., Professor, Eskisehir Technical University, Faculty of Science, 26470 Eskisehir, Turkey, e-mail: kguseynov@eskisehir.edu.tr
Cite this article as: N. Huseyin, A. Huseyin, Kh.G. Guseinov. On the Properties of the Set of Trajectories of Nonlinear Control Systems with Integral Constraints on the Control Functions. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, vol. 28, no. 3, pp. 274–284.
Русский
Н. Гусейин, А. Гусейин, Х.Г. Гусейнов. О свойствах множества траекторий нелинейной управляемой системы с интегральными ограничениями на управляющие функции
Исследуются управляемые системы, описываемые нелинейными дифференциальными уравнениями с интегральными ограничениями на управляющие функции. Допустимые управляющие функции выбираются из замкнутого шара пространства $L_p$, $p\in (1,\infty]$, радиуса $r$ с центром в начале координат. Доказано, что множество траекторий системы непрерывно при $p=\infty$ в псевдометрике Хаусдорфа. Показано, что каждая траектория робастна по отношению к быстрому и полному расходованию оставшегося управляющего ресурса, из чего следует, что для достижения желаемого результата целесообразно расходовать имеющийся ресурс управления малыми порциями. Благодаря этому удается доказать, что каждая траектория может быть аппроксимирована траекторией, соответствующей полному расходованию ресурса управления.
Ключевые слова: нелинейная управляемая система, множество траекторий, интегральное ограничение, геометрическое ограничение, непрерывность по Хаусдорфу, робастность
