Second-Order Necessary Optimality Conditions for the Mayer Problem Subject to a General Control Constraint

This paper is devoted to second-order necessary optimality conditions for the Mayer optimal control problem with an arbitrary closed control set U subset of R-m. Admissible controls are supposed to be measurable and essentially bounded. Using second order tangents to U, we first show that if (u) over bar(.) is an optimal control, then an associated quadratic functional should be nonnegative for all elements in the second order jets to U along (u) over bar(.). Then we specify the obtained results in the case when U is given by a finite number of C-2-smooth inequalities with positively independent gradients of active constraints. The novelty of our approach is due, on one hand, to the arbitrariness of U. On the other hand, the proofs we propose are quite straightforward and do not use embedding of the problem into a class of infinite dimensional mathematical programming type problems. As an application we derive new second-order necessary conditions for a free end-time optimal control problem in the case when an optimal control is piecewise Lipschitz.