The problem of simple group pursuit with phase constraints in time scales

In the finite-dimensional Euclidean space $\mathbb R^k,$ the problem of pursuit of one evader by a group of pursuers with equal opportunities for all participants is considered, which is described in a given time scale $T$ by a system of the form
$$z_i^{\Delta} = u_i - v,$$
where $f^{\Delta}$ is the $\Delta$-derivative of the function $f$ in the time scale $T$. The set of admissible controls is a ball of unit radius with the center at the origin. Terminal sets are the coordinate origin. Additionally, it is assumed that the evader does not leave the convex polyhedral set with a nonempty interior during the game. Sufficient conditions for the solvability of the pursuit and evasion problems are obtained. In the research, the method of resolving functions is used as the basic one.