In this paper the control system with integral constraint on the control functions is studied where the behavior of the system by the Urysohn type integral equation is described. The admissible control functions are chosen from the closed ball of the space $L_p ([a,b];R^m )$ $(p>1)$ centered at the origin with radius $r$. Dependence of the set of trajectories on r and p is investigated. It is proved that the set of trajectories is Lipschitz continuous with respect to r and continuous with respect to $p$. The robustness of the trajectory with respect to the fast consumption of the remaining control resource is established.