The Goh necessary optimality conditions for the Mayer problem with control constraints

The well known Goh second order necessary optimality conditions in optimal control theory concern singular optimal controls taking values in the interior of a set of controls U. In this paper we investigate these conditions for the Mayer problem when U is a convex polytope or a closed subset of class C² for an integrable optimal control u̅(·) that may take values in the boundary of U. This is indeed a frequent situation in optimal control and for this reason the understanding of this issue is crucial for the theory of second order optimality conditions. Applying the Goh transformation we derive necessary conditions on tangent subspace to U at u̅(t) for almost all t´s. In the presence of an endpoint constraint, if the Mayer problem is calm, then similar second order necessary optimality conditions are satisfied whenever the maximum principle is abnormal. If it is normal, then analogous results hold true on some smaller subspaces.