  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
smooth 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.
