Second-order optimality conditions for a Bolza problem with mixed constraints ?

For an optimal control problem with fixed initial state and constraints on admissible controls we propose new second-order necessary optimality conditions using second-order tangents to the set of controls. In the absence of constraints on controls, our conditions are stronger than those of Gilbert and Bernstein (1983). Then, we investigate optimal control problems with fixed initial state and mixed statecontrol equality constraints. Reducing these problems to problems without mixed constraints, we provide firstand second-order necessary optimality conditions. Our results improve some existing results (see Milyutin and Osmolovskii (1998); Maurer and Osmolovskii (2003); Osmolovskii (2004)). In particular, we do not make the assumption that the optimal control is piecewise continuous.