Localization method and its applications

Conditions for asymptotic stability and global asymptotic stability of equilibrium points of nonlinear time-invariant systems are obtained in terms of compact invariant sets and positively invariant sets. To verify these conditions the functional localization method of compact invariant sets is proposed. The obtained results can be applied to investigate the asymptotic stability in critical cases, and to find the Lyapunov function if the equilibrium point is (globally) asymptotically stable.