On the p-integrable trajectories of the nonlinear control system described by Urysohn type integral equation

The control systems described by the Urysohn-type integral equations and integral constraints on the control functions are considered. The functions from the closed ball of the space $L_p$, $p>1$, with radius $r$, are chosen as admissible control functions. The trajectory of the system is defined as a $p$-integrable function, satisfying the system’s equation almost everywhere. The boundedness and path-connectedness of the set of $p$-integrable trajectories are discussed. It is illustrated that the set of trajectories, in general, is not a closed subset of the space $L_p$. The robustness of a trajectory with respect to the fast consumption of the remaining control resource is established, and it is proved that every trajectory of the system can be approximated by the trajectory obtained by the full consumption of the control resource.