Mixed constrained control problems

Necessary optimality conditions are derived in the form of a weak maximum principle for optimal control problems with mixed state-control equality and inequality constraints. In contrast to previous work these conditions hold when the Jacobian of the active constraints, with respect to the unconstrained control variable, has full rank. A feature of these conditions is that they are stated in terms of a joint Clarke subdifferential. Furthermore the use of the joint subdifferential gives sufficiency for nonsmooth, normal, linear convex problems. The main point of interest is not only the full rank condition assumption but also the nature of the analysis employed in this paper. A key element is the removal of the constraints and application of Ekeland's variational principle.