Properties of the Set of L∞L∞ Trajectories of the Control Systems With Limited Control Resources

In this paper, the set of trajectories of the control system described by Urysohn type integral equation is considered. It is assumed that the system is nonlinear with respect to the state vector and affine with respect to the control vector. The closed ball of the space , is chosen as the set of admissible control functions. The trajectory of the system is defined as multivariable Lebesgue measurable function from the space that satisfies the system's equation almost everywhere. Boundedness of the set of trajectories is shown, and it is proved that every sequence of trajectories has a subsequence that converges almost everywhere to a system's trajectory. Existence of the optimal process in the optimal control problem with linear quality functional is presented. It is shown that every trajectory is robust with respect to the fast consumption of the remaining control resource and the set of trajectories as a set valued map depending on is continuous with respect to in the Hausdorff pseudometric generated by the norm of the space .