On two problems of pursuit of a group of evaders in differential games with fractional derivatives

Abstract

In a finite-dimensional Euclidean space, the problem of pursuit of a group of evaders by a group of pursuers is considered, described by a system of the form \begin{gather*} D^{(\alpha)}x_i = a_i x_i + u_i, \ u_i \in U_i, \quad D^{(\alpha)}y_j = b_jy_j + v, \ v\in V, \end{gather*} where $D^{(\alpha)}f$ is the Caputo derivative of order $\alpha$ of the function $f$. The sets of admissible controls $U_i, V$ are convex compacts, $a_i, b_j$ are real numbers. The terminal sets are convex compacts. Sufficient conditions for the solvability of the pursuit problems are obtained. In the study, the method of resolving functions is used as the basic one. It is shown that such a conflict situation with equal opportunities for all participants is possible, in which one pursuer catches all the evaders.