  Higher order necessary conditions for a minimizer of an optimal control
problem are generally obtained for systems whose dynamics is at least
continuously differentiable in the state variable. Here, by making use of the
notion of set-valued Lie bracket introduced in "Set-valued differentials and a
nonsmooth version of Chow-Rashevski's theorem" by F. Rampazzo and H.J.Sussmann
and extended in "Iterated Lie brackets for nonsmooth vector fields" by E.
Feleqi and F.Rampazzo , we obtain Goh and Legendre-Clebsh type conditions for a
control affine system with Lipschitz continuous dynamics. In order to manage
the simultaneous lack of smoothness of the adjoint equation and of the Lie
bracket-like variations, we will exploit the notion of Quasi Differential
Quotient, introduced in "A geometrically based criterion to avoid infimum-gaps
in Optimal Control" by M. Palladino and F.Rampazzo. We finally exhibit an
example where the established higher order condition is capable to rule out the
optimality of a control verifying a first order Maximum Principle.
