A problem of simple group pursuit with possible dynamical disturbance in dynamics and phase constraints

In finite-dimensional Euclidian space, we treat the problem of pursuit of one evader by a group of pursuers, which is described by a system of the form
$$ \dot z_i = a_i(t) u_i - v, \quad u_i\in U_i, \quad v\in V, $$
where the functions $\alpha_i(t)$ are equal to $1$ for all $t$, except for a certain interval of a given length, on which they are equal to zero (to each pursuer there corresponds its own interval). This fact can be interpreted in such a way that each of the pursuers has a possible failure of the control device at any previously unknown moment in time, and the length of the time interval needed to fix the failure is given, while in the process of fixing the failure the pursuers have no possibility to carry out a capture. The target sets are convex compact sets, and the evader does not leave the bounds of the convex polyhedral set. We obtain sufficient conditions for solvability of the pursuit problem.