An optimal control problem for the systems with integral boundary conditions

In this paper, we consider an optimal control problem with a «pure», integral boundary condition. The Green’s function is constructed. Using contracting Banach mappings, a sufficient condition for the existence and uniqueness of a solution to one class of integral boundary value problems for fixed admissible controls is established. Using the functional increment method, the Pontryagin‘s maximum principle is proved. The first and second variations of the functional are calculated. Further, various necessary conditions for optimality of the second order are obtained by using variations of controls.