Stability robustness in the presence of exponentially unstable isolated equilibria

This note studies nonlinear systems evolving on manifolds with a finite number of asymptotically stable equilibria and a Lyapunov function which strictly decreases outside equilibrium points. If the linearizations at unstable equilibria have at least one eigenvalue with positive real part, then almost global asymptotic stability turns out to be robust with respect to sufficiently small disturbances in the L_∞ norm. Applications of this result are shown in the study of almost global Input-to-State stability.