On the full rankness condition for mixed constrained optimal control problems

Optimal control problems with mixed constraints in the form of equalities and inequalities and possibly nonsmooth data are considered. We report on two weak maximum principles recently obtained for such problems. These two results hold when a certain matrix is assumed to be of full rank. The full rank of the matrix considered in the first case implies the full rank of the matrix considered in the second case, but, as we show through an example, the opposite implication is not valid. Although the set of optimality conditions obtained are essentially the same, the replacement of one matrix by another drastically changes the proofs. We discuss such feature and outlines of proofs of the above mentioned results are provided. Of special interest is the fact that the optimality conditions we report on are stated in terms of a joint Clarke subdifferential of the Hamiltonian. Noteworthy, the use of the joint subdifferential provides sufficiency for nonsmooth, normal, linear convex problems. Copyright (C) 2003 IFAC.