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

The control system with integralconstraint on the controls is studied, where the behavior of the system by a Urysohn type integral equation is described. It is assumed thatthe system is nonlinear with respect to the state vector, affine with respect to the control vector. The closed ball ofthe space $L_p(E;\mathbb{R}^m)$ $(p>1)$ with radius $r$ and centered at theorigin, is chosen as the set of admissible control functions, where $E\subset \mathbb{R}^k$ is a compact set. Itis proved that the set of trajectories generated by all admissible control functions is a compact subset of the space of continuous functions.