A smooth Lyapunov function from a class- ${\mathcal{KL}}$ estimate involving two positive semidefinite functions

We consider dierential inclusions where a positive semidenite function of the solutions satises a class-KL estimate in terms of time and a second positive semidenite function of the initial condition. We show that a smooth converse Lyapunov function, i.e., one whose derivative along solutions 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 dierential inclusion. It remains an open question whether all class-KL estimates are robust. One sucient condition for robustness is that the original dierential inclusion is locally Lipschitz. Another sucient condition is that the two positive semidenite functions agree and a backward completability condition holds. These special cases unify and generalize many results on converse Lyapunov theorems for dierential equations and dierential inclusions that have appeared in the literature.