MSC: 93C10, 45G15
DOI: 10.21538/0134-4889-2021-27-3-263-270
Full text
In this paper a control system described by an Urysohn type integral equation with an integral constraint on the control functions is studied. It is assumed that the system is nonlinear with respect to the state vector and is affine with respect to the control vector. The control functions are chosen from a closed ball of the space $L_p$ $(p>1)$ with radius $r$. It is proved that the set of trajectories of the control system generated by all admissible control functions is Lipschitz continuous with respect to $r$ and is continuous with respect to $p$ as a set valued map. It is shown that the system's trajectory is robust with respect to the full consumption of the remaining control resource and every trajectory can be approximated by a trajectory generated by the control function with full control resource consumption.
Keywords: integral equation, control system, integral constraint, set of trajectories, robustness
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Received March 25, 2021
Revised May 31, 2021
Accepted June 15, 2021
Nesir Huseyin, Doctor of Philosophy, Associate Professor, Faculty of Education, Sivas Cumhuriyet University, Turkey, e-mail: nhuseyin@cumhuriyet.edu.tr
Anar Huseyin, Doctor of Philosophy, Associate Professor, Faculty of Science, Sivas Cumhuriyet University, Turkey, e-mail: ahuseyin@cumhuriyet.edu.tr
Khalik Garakishi oglu Guseinov, Doctor of Physical and Mathematical Sciences, Professor, Faculty of Science, Eskisehir Technical University, Turkey, e-mail: kguseynov@eskisehir.edu.tr
Cite this article as: N. Huseyin, A. Huseyin, Kh.G. Guseinov. On the Robustness Property of the Control System Described by Urysohn Type Integral Equation, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2021, vol. 27, no. 3, pp. 263–270.
Русский
Н. Гусейин, А. Гусейин, Х.Г. Гусейнов. О свойстве робастности управляемой системы, описываемой интегральным уравнением типа Урысона
В данной работе исследуется управляемая система, описываемая интегральным уравнением типа Урысона с интегральным ограничением на управляющие функции. Предполагается, что система нелинейна по фазовому вектору и аффинна по управляющему вектору. Управления выбираются из замкнутого шара пространства $L_p$ $(p>1)$ радиуса $r$ с центром в начале координат. Доказано, что множество траекторий управляемой системы, отвечающих всем допустимым управлениям, липшицево по $r$ и непрерывно по $p$ как многозначное отображение. Показано, что траектория системы робастна относительно полного использования оставшегося ресурса управления и любую траекторию системы можно аппроксимировать траекторией, соответствующей управлению с полным использованием ресурса.
Ключевые слова: интегральное уравнение, управляемая система, интегральное ограничение, множество траекторий, робастность
