On second-order necessary optimality conditions in the classical sense for systems with nonlocal conditions

We consider an optimal control problem in which the state of a system is described by control systems of ordinary differential equations with two-point boundary conditions. The admissible controls are chosen in the class of bounded measurable functions. We derive a formula for the increment of a second-order functional. On the basis of control variations, we obtain necessary optimality conditions for singular controls in the classical sense.