Group pursuit in recurrent Pontryagin example

A non-stationary differential game (a generalized example of L. S. Pontryagin) with $n$ pursuers and one evader is considered in the space $\mathbb R^k$ ($k\geqslant2$). All players have equal dynamic and inertial capabilities. The game is described by a system of the form
\begin{gather*} Lz_i=z_i^{(l)}+a_1(t)z_i^{(l-1)}+\dots+a_l(t)z_i=u_i-v,\quad u_i,v\in V,\\ z_i^{(s)}(t_0)=z_{is}^0,\qquad i=1,2,\ldots,n,\quad s=0,1,\ldots,l-1. \end{gather*}
The set $V$ of admissible player controls is strictly convex compact set with smooth boundary, $a_1(t),\dots,a_l(t)$ are continuous on $[t_0,\infty)$ functions, the terminal sets are the origin of coordinates. Pursuers use quasi-strategies. It is assumed that functions $\xi_i(t)$ being the solution of Cauchy problem
$$ Lz_i=0,\quad z_i^{(s)}(t_0)=z_{is}^0, $$
are recurrent. Properties of recurrent functions are given. In terms of initial positions and game parameters the sufficient conditions of the pursuit problem solvability are obtained. The proof is carried out using the method of resolving functions. An example illustrating the obtained conditions is given.