Application of the method of Lyapunov periodic functions

The paper is concerned with asymptotic behavior of continuous and discrete phase control systems involving periodic differentiable nonlinear vector functions and featuring nonunique equilibrium state. Two stability problems are examined in sequence: the problem of global asymptotics and the problem on the number of cycle slippings. A conventional tool in attacking such problems is the second Lyapunov method. However, Lyapunov functions of the kind "quadratic form" or "quadratic form plus the integral of nonlinear function," which are conventional in the control theory, are not capable of solving these problems for the class of systems under consideration. As a result, several new methods were put forward in the 1960s and 1970s to deal with phase control systems in the framework of the second Lyapunov method. In this work, one of these methods is exploited; namely, the method of Lyapunov periodic functions. Extensions of well-known Lyapunov periodic functions and sequences are proposed permitting us to refine estimates of regions of global asymptotics in the space of parameters of a phase system. Multiparameter frequency criteria for global asymptotics are formulated; they allow for improvement of the aforementioned estimates. The frequency criteria obtained are used as well to establish estimates for the number of cycle slippings. © 2011 Allerton Press, Inc.