Unmaximized necessary conditions for constrained control problems

Abstract

New first-order necessary conditions of optimality for control problems with mixed state-control and pure state constraints are derived. In contrast to known results these conditions hold when the Jacobian of the active state-control constraints with respect to the control has full rank. A crucial feature of these conditions is that they are stated in terms of a joint subdifferential and do not involve the maximization of the Hamiltonian. The main novelty of the result is precisely the ability to address state-control and pure state constraints, generalizing previously proved results. The conditions developed are, in some cases, stronger than the standard nonsmooth maximum principle, since they can reduce the set of candidates to minimizers.