Pointwise Second-Order Necessary Optimality Conditions for the Mayer Problem with Control Constraints

This paper is devoted to second order necessary optimality conditions for the Mayer optimal control problem when the control set U is a closed subset of R. We show that, in the absence of endpoint constraints, if an optimal control ū(·) is singular and integrable, then for almost every t such that ū(t) is in the interior of U , both the Goh and a generalized LegendreClebsch conditions hold true. Moreover, when the control set is a convex polytope, similar conditions are verified on the tangent subspace to U at ū(t) for almost all t’s such that ū(t) lies on the boundary ∂U of U . The same conditions are valid also for U having a smooth boundary at every t where ū(·) is singular, locally Lipschitz and ū(t) ∈ ∂U. In the presence of a smooth endpoint constraint, these second order necessary optimality conditions are satisfied whenever the Mayer problem is calm and the maximum principle is abnormal. If it is normal, then analogous results hold true on some smaller subspaces.