Strengthening the conditions of Clarke and Smirnov for convex-valued differential inclusions

Assume that a Lipschitz continuous differential inclusion with convex images and locally compact graph is fixed on a certain time interval. For trajectories of this inclusion the problem of the minimization of a smooth end-point function is considered under smooth end-point constraints of equality and inequality types. This problem is approximated by a sequence of smooth optimal control problems with regular mixed constraints, which are treated using the maximum principle obtained earlier by the author in conjunction with Dubovitskii. Passing to the limit in the conditions of the maximum principle one obtains necessary conditions for strong minimality in the initial problem which refine the well-known conditions of Clarke and Smirnov.