A smooth Lyapunov function from a class-KL estimate involving two positivesemidefinite functions

We consider differential inclusions where a positive semidefinite function of the solutions satisfies a class-KL estimate in terms of time and a secon d positive semidefinite function of the initial condition. We show that a s mooth converse Lyapunov function, i.e., one whose derivative along solution s can be used to establish the class-KL estimate, exists if and only if the class-KL estimate is robust, i.e., it holds for a larger, perturbed differ ential inclusion. It remains an open question whether all class-KL estimate s are robust. One sufficient condition for robustness is that the original differential inclusion is locally Lipschitz. Another sufficient condition i s that the two positive semidefinite functions agree and a backward complet ability condition holds. These special cases unify and generalize many resu lts on converse Lyapunov theorems for differential equations and differenti al inclusions that have appeared in the literature.