On a group pursuit problem on time scales

Abstract

In a finite-dimensional Euclidean space $\mathbb R^k$, we consider a linear problem of pursuit of one evader by a group of pursuers, which is described on the given time scale $\mathbb{T}$ by equations of the form \begin{gather*} z_i^{\Delta} = a z_i + u_i - v, \end{gather*} where $z_i^{\Delta}$ is the $\Delta$-derivative of the functions $z_i$ on the time scale $\mathbb{T}$, $a$ is an arbitrary number not equal to zero. The set of admissible controls for each participant is a unit ball centered at the origin, the terminal sets are given convex compact sets in $\mathbb R^k$. The pursuers act according to the counter-strategies based on the information about the initial positions and the evader control history. In terms of initial positions and game parameters, a sufficient capture condition has been obtained. For the case of setting the time scale in the form $\mathbb T = \{ \tau k \mid k \in \mathbb Z,\ \tau \in \mathbb R,\ \tau >0\}$ sufficient pursuit and evasion problems solvability conditions have been found. In the study, in both cases, the resolving function method is used as basic one.