Almost Lyapunov functions for nonlinear systems

We study convergence of nonlinear systems in the presence of an “almost Lyapunov” function which, unlike the classical Lyapunov function, is allowed to be nondecreasing – and even increasing – on a nontrivial subset of the phase space. Under the assumption that the vector field is free of singular points (away from the origin) and that the subset where the Lyapunov function does not decrease is sufficiently small, we prove that solutions approach a small neighborhood of the origin. A nontrivial example where this theorem applies is constructed.