Almost Lyapunov functions

An autonomous, time-invariant dynamical system that verifies the classical Lyapunov criterion is globally asymptotically stable for the case when the Lyapunov function is strictly decreasing along the solution. We define an almost Lyapunov function to be a candidate Lyapunov function whose derivative with respect to time is of unknown sign in a set of small measure and negative elsewhere. We investigate the convergence of a system associated with an almost Lyapunov function. We show that for an unknown set of small enough measure, all the trajectories converge to a ball around the equilibrium point, under standard condition of Lipschitz continuity over the system and the derivative of the almost Lyapunov function