Leonov’s nonlocal reduction technique for nonlinear integro-differential equations

Starting from pioneering works by Lur’e, Popov and Zames, global stability theory for nonlinear control systems has been primarily focused on systems with only one equilibrium. Global stability criteria for other kinds of attractors (such as e.g. infinite sets of equilibria) are not well studied and typically require special tools, primarily based on the Lyapunov method. Analysis of stability becomes especially complicated for infinite-dimensional dynamical systems with multiple equilibria, e.g. systems described by delay or more general convolutionary equations. In this paper, we propose novel stability criteria for infinite-dimensional systems with periodic nonlinearities, which have infinite sets of equilibria and describe dynamics of phase-locked loops and other synchronization circuits. Our method combines Leonov’s nonlocal reduction technique with the idea of Popov’s “integral indices” and allows to obtain new frequency-domain conditions, ensuring the convergence of every solution to one of the equilibria points.