On the validity of the transversality condition for different concepts of tangent cone to a set

Abstract

In the nonsmooth versions of the Pontryagin Maximum Principle, the transversality condition involves a normal cone to the terminal set. General versions of the principle for highly non-smooth systems have been proved by separation methods for cases that include, for example, a reference vector field which is classically differentiable along the reference trajectory but not Lipschitz. In these versions, the notion of normal cone used is that of the polar of a Boltyanskii approximating cone. Using a recent result of A. Bressan, we prove that these versions can fail to be true if the Clarke normal cone (and, a fortiori, any smaller normal cone, such as the Mordukhovich cone) is used instead. The key fact is A. Bressan's recent example of two closed sets that intersect at a point p and are such that (a) one of the sets has a Boltyanskii approximating cone C1 at p, (b) the other set has a Clarke tangent cone C2 at p, and (c) the cones C1 and C2 are strongly transversal, but (d) the sets only intersect at p.