Compactness of the Set of Trajectories of the Control System Described by a Urysohn Type Integral Equation

The control system with integral constraint on the controls is studied, wherethe behavior of the system by a Urysohn type integral equation is described. Itis assumed that the system is nonlinear with respect to the state vector, affinewith respect to the control vector. The closed ball of the spaceLp(E;Rm) (p >1) with radiusrand centered at the origin, is chosen as the set of admissiblecontrol functions, whereE⊂Rkis a compact set. It is proved that the set oftrajectories generated by all admissible control functions is a compact subsetof the space of continuous functions