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

Goh and Legendre-Clebsh conditions for nonsmooth control systems