Approximation of the set of trajectories of a control system described by the Urysohn integral equation

The approximation of the set of trajectories of a control system described by the Urysohn integral equation is considered. The closed ball of the space $L_p([a,b];\mathbb{R}^m)$ $(p>1)$ of radius $r$ centered at the origin is chosen as the set of admissible controls. This set is replaced by a set of control functions, which consists of a finite number of controls and generates a finite number of trajectories. An accuracy estimate is obtained for the Hausdorff distance between the set of trajectories and the set consisting of a finite number of trajectories.