The critical tangent cone in second-order conditions for optimal control

In this paper the notion of critical tangent cone CT(x/Q) is introduced. Wh en Q is closed, convex with nonempty interior, x is an element of Q then th e nonemptiness of the Dubovitskii-Milyutin set of second-order admissible v ariations, V(x, d/Q), is characterized by the condition d is an element of CT(x/Q). More verifiable characterization is obtained for the cases where Q is the set of continuous or L-infinity selections of a certain set-valued map. In the latter case, a strong normality condition in terms of CT(x(t)/Q (t)) is defined in order that the Lagrange multiplier corresponding to the L-infinity-selections set be represented via integrable functions. Finally these results are applied to a general optimal control problem and second-o rder optimality conditions are derived in terms of the original data.