MSC: 93C23, 93C35, 45G15
DOI: 10.21538/0134-4889-2026-32-3-71-84
An approximation of a precompact set of trajectories of a control system described by the Urysohn type integral equation and an integral constraint on the control functions is considered. A trajectory of the system is defined as a multivariable $p$-integrable ($p>1$) function that satisfies the system's equation almost everywhere. Admissible control functions are chosen from a closed ball of the space $L_p$, $p>1$, with a radius $r$ centered at the origin. Step by step, the set of admissible control functions is replaced by a set that consists of a finite number of piecewise-constant control functions that generate a finite number of trajectories. It is proved that, with appropriate definitions of the discretization parameters, a set consisting of a finite number of trajectories is an inner approximation of the set of trajectories of the system. The paper also discusses an approximation of the set of trajectories with mixed constraints on the control functions.
Keywords: nonlinear control system, Urysohn integral equation, integrable trajectory, integral constraint, mixed constraint, approximation
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Received December 21, 2025
Revised May 15, 2026
Accepted May 18, 2026
Anar Huseyin, Prof. Dr., Sivas Cumhuriyet University, Faculty of Science, Department of Statistics and Computer Sciences, Sivas, 58140 Turkey, e-mail: ahuseyin@cumhuriyet.edu.tr
Nesir Huseyin, Prof. Dr., Sivas Cumhuriyet University, Faculty of Education, Department of Mathematics and Science Education, Sivas, 58140 Turkey, e-mail: nhuseyin@cumhuriyet.edu.tr
Khalik G. Guseinov, Dr. Phys.-Math. Sci., Prof., Eskisehir Technical University, Faculty of Science, Department of Mathematics, Eskisehir, 26470 Turkey, e-mail: k.g.guseinov@gmail.com
Cite this article as: A. Huseyin, N. Huseyin, Kh.G. Guseinov. Approximation of the set of integrable trajectories of a nonlinear control system with an $L_p$-norm constraint on the control functions. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2026, vol 32, no. 3, pp. 71–84
Русский
А. Гусейин, Н. Гусейин, Х.Г. Гусейнов. Аппроксимация множества интегрируемых траекторий нелинейной управляемой системы с $L_p$ норм ограничением на функции управления
Рассматривается аппроксимация предкомпактного множества траекторий управляемой системы, описываемой интегральным уравнением типа Урысона и интегральным ограничением на функции управления. Траектория системы определяется как многомерная $p$-интегрируемая ($p>1$) функция, удовлетворяющая уравнению системы почти всюду. Допустимые функции управления выбираются из замкнутого шара пространства $L_p$, $p>1$, с радиусом $r$ и центром в начале координат. Шаг за шагом, множество допустимых функций управления заменяется множеством, состоящим из конечного числа кусочно-постоянных функций управления, которые порождают конечное число траекторий. Доказывается, что при соответствующих определениях параметров дискретизации, множество, состоящее из конечного числа траекторий, является внутренней аппроксимацией множества траекторий системы.
Ключевые слова: нелинейная система управления, интегральное уравнение Урысона, интегрируемая траектория, интегральное ограничение, аппроксимация
