Compactness of the set of trajectories of the controllable system described by an affine integral equation

In this paper the control system with integral constraint on the controls is studied. It is assumed that the behavior of the system is described by an affine Volterra integral equation which is nonlinear with respect to the state vector and is linear with respect to the control vector. The closed ball of the space Lp (p>1) with radius μ0 and centered at the origin, is chosen as the set of admissible control functions. It is proved that the set of trajectories is a bounded, closed, precompact and hence compact subset of the space of continuous functions.