For a given closed target we embed the dissipative relation that defines a control Lyapunov function in a more general differential inequality involving Hamiltonians built from iterated Lie brackets. The solutions of the resulting extended relation, here called degree-k control Lyapunov functions (k >= 1), turn out to be still sufficient for the system to be globally asymptotically controllable to the target. Furthermore, we work out some examples where no standard (i.e., degree-1) smooth control Lyapunov functions exist while a C-infinity degree-k control Lyapunov function does exist for some k > 1. The extension is performed under very weak regularity assumptions on the system, to the point that, for instance, (set-valued) Lie brackets of locally Lipschitz vector fields are considered as well.