Control engineering applications of V. I. Zubov's construction procedure for Lyapunov functions

The determination of a stability domain of a control system, the motion of which is described by nonlinear differential equations, is often the object of intensive experimental and theoretical attack. This paper, partly tutorial and partly a presentation of new results, describes a method for obtaining a solution to this problem proposed recently by the Russian mathematician, V. I. Zubov. The tutorial part outlines the fundamental principles of V. I. Zubov's procedure for constructing Lyapunov functions for non-linear systems. If the construction problem can be solved, it leads to a Lyapunov function which uniquely defines the exact boundary of the stability region. For the application of the method, several simple examples are treated in which the exact stability region is found in analytic closed form. Since the construction procedure requires the solution of a linear partial differential equation, there are many cases for which an exact analytic solution is not possible. In some of these cases, however, it is possible to construct an approximate series solution which is always at least as good as the usual quadratic form Lyapunov function. The series construction procedure has been programmed (in IBM 7070 FORTRAN language) for a broad class of differential equations of the second order. A simple example solved by the digital computer program is described.