A problem of pursuit of one evader by a group of pursuers is considered in a finite-dimensional Euclidean space. The dynamics is described by the system
$$ D^{(\alpha_i)}z_i = A_iz_i + B_iu_i - C_iv, \quad u_i\in U_i,\quad v\in V, $$
where $D^{(\alpha)}f$ is the Caputo derivative of order $\alpha$ of a function $f$. The sets of admissible controls of the players are convex and compact. The terminal set consists of cylindrical sets $M_i$ of the form $M_i = M_i^1 + M_i^2$, where $M_i^1$ is a linear subspace of the phase space and $M_i^2$ is a convex compact set from the orthogonal complement of $M_i^1$. We propose two approaches to solving the problem, which ensure the termination of the game in a certain guaranteed time in the class of quasi-strategies. In the first approach, the pursuers construct their controls so that the terminal sets "cover" the evader's uncertainty region. In the second approach, the pursuers construct their controls using resolving functions. The theoretical results are illustrated by model examples.
Keywords: differential game, group pursuit, pursuer, evader, fractional derivative
Received May 30, 2022
Revised July 7, 2022
Accepted July 11, 2022
Funding Agency: This work was supported by the Russian Science Foundation (project no. 21-71-10070).
Alena Igorevna Machtakova, doctoral student, Udmurt State University, Izhevsk, 426034 Russia; Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: bichurina.alyona@yandex.ru
Nikolai Nikandrovich Petrov, Dr. Phys.-Math. Sci, Prof., Udmurt State University, Izhevsk, 426034, Russia, Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: kma3@list.ru
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Cite this article as: A.I. Machtakova, N.N. Petrov. On a linear group pursuit problem with fractional derivatives. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, vol. 28, no. 3, pp. 129–141; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2022, Vol. 319, Suppl. 1, pp. S175–S187.
