Nonsmooth control-Lyapunov functions

It is shown that the existence of a continuous control-Lyapunov function (CLF) is necessary and sufficient for null asymptotic controllability of nonlinear finite-dimensional control systems. The CLF condition is expressed in terms of a concept of generalized derivative that has been studied in set-valued analysis and the theory of differential inclusions with various names such as “upper contingent derivative”. This result generalizes to the nonsmooth case the theorem of Artstein (1983) relating closed-loop feedback stabilization to smooth CLF´s. It relies on viability theory as well as optimal control techniques. A “nonstrict” version of the results, analogous to the LaSalle invariance principle, is also provided