Existence of the optimal 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 L_p (p >1) with gives radius μ₀ and centered at the origin, is chosen as the set of admissible control functions. Compactness of the set of trajectories and existence of the optimal trajectories in the optimal control problems with semi-continuous cost functionals are proved.