Leonov’s method of nonlocal reduction for pointwise stability of phase systems

In this paper we go on with the analysis of the asymptotic behavior of Lur’e–type systems with periodic nonlinearities and infinite sets of equilibria. It is well known by now that this class of systems can not be efficiently investigated by the second Lyapunov method with the standard Lur’e–Postnikov function ("a quadratic form plus an integral of the nonlinearity"). So several new methods have been elaborated within the framework of Lyapunov direct method. The nonlocal reduction technique proposed by G.A. Leonov in the 1980s is based on the comparison principle. The feedback system is reduced to a low-order system with the same nonlinearity and known asymptotic behavior. Its trajectories are injected into Lyapunov function of the original system. In this paper we develop the method of nonlocal reduction. We propose a new Lyapunov–type function which involves both the trajectories of the comparison system and a modified Lur’e–Postnikov function. As a result a new frequency–algebraic criterion ensuring the convergence of every solution to some equilibrium point is obtained.