MSC: 93B99
https://doi.org/10.21538/0134-4889-2025-31-3-fon-02
The work was performed as part of research conducted in the Ural Mathematical Center with the financial support of the Ministry of Science and Higher Education of the Russian Federation (Agreement number 075-02-2025-1549).
We consider a controlled linear differential equation. The controller must transfer the initial state $x_0$ of the equation to a given final state $x_T$. This process is followed by the observer, who tries to determine $x_T$ but does not know the state vector of the equation and obtains information via the vector $y(t)$ connected with $x(t)$. With the aid of the signal $y(t)$, the observer can determine an information set containing $x_T$. In the case of special constraints for controls (or disturbances from the point of view of the observer), the information set becomes the ellipsoid, the parameters of which are described by the system of differential equations. In the game, the controller, who is the main player, endeavors to accomplish its task and maximize the information set simultaneously. An example is considered.
Keywords: guaranteed estimation, information set, reachable set, observation control.
Б.И. Ананьев. Задача управления наблюдением для дифференциальных уравнений
Рассматривается управляемое линейное дифференциальное уравнение. Управляющее лицо должно переместить начальное состояние $x_0$ уравнения в фиксированное конечное состояние $x_T$. Этот процесс контролируется наблюдателем, который пытается определить $x_T$, но не знает фазовый вектор уравнения и получает информацию от вектора $y(t)$, связанного с $x(t)$. С помощью сигнала $y(t)$ наблюдатель может определить информационное множество, содержащее $x_T$. В случае специальных ограничений на управления (или возмущений с точки зрения наблюдателя) информационное множество становится эллипсоидом, параметры которого описываются системой дифференциальных уравнений. Управляющее лицо, которое является основным, пытается выполнить свою задачу и одновременно максимизировать размер информационного множества. Рассмотрен пример.
Ключевые слова: гарантированное оценивание, информационное множество, множество достижимости, управление наблюдением.
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Received April 25, 2025
Revised May 12, 2025
Accepted May 12, 2025
Published online June 19, 2025
Funding Agency: The work was performed as part of research conducted in the Ural Mathematical Center with the financial support of the Ministry of Science and Higher Education of the Russian Federation (Agreement number 075-02-2025-1549).
Boris Ivanovich Ananyev, Dr. Phys.-Math. Sci., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: abi@imm.uran.ru
Cite this article as: B.I. Ananyev. Observation control problem for differential equations. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2025, vol. 31, no. 3, pp. 36–46.
