Nonsmooth functions and uniform limits of smooth control Lyapunov functions

When is a (non-smooth) function the limit of a sequence of smooth (continuously differentiable) control Lyapunov functions? It is known that a "non-smooth control Lyapunov function in the sense of the Clarke (convex) gradient" is indeed the limit of a sequence of smooth control Lyapunov functions; we show that the converse is not true by exhibiting a counter example. We also give a condition under which a function cannot be the limit of a sequence of smooth control Lyapunov functions. Of course, a (non smooth) function that satisfies this condition cannot either be a control Lyapunov function in the sense of the Clarke gradient.