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 m. We show that, in the absence of endpoint constraints, if an optimal control (Formula Presentd.) is singular and integrable, then for almost every t such that (Formula Presentd.) is in the interior of U, both the Goh and a generalized Legendre-Clebsch condition hold true. Moreover, when the control set is a convex polytope, similar conditions are verified on the tangent subspace to U at (Formula Presentd.) for almost all t's such that (Formula Presentd.) lies on the boundary ∂U of U. The same conditions are valid also for U having a smooth boundary at every t where (Formula Presentd.) is singular and locally Lipschitz and (Formula Presentd.). 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. © 2013 Society for Industrial and Applied Mathematics.