A Format Method for Generating Liapunov Functions

Abstract

A method is developed for generating Liapunov functions, V, which have the property that, along the trajectories of a system under consideration, the scalar functions V̇ or (V̇ − Vt ) take on a preassigned or desired form. The method applies to both autonomous and nonautonomous systems and complements and extends other known methods and techniques for generating Liapunov functions. The method is called a format method since it is based upon a fundamental vector-matrix equation or format, v = [D + P]f , which mathematically represents every vector v which satisfies the scalar product v ·f = (V̇ − Vt ). The method is readily applied to very general classes of systems as well as to special and particular systems. The format method is illustrated by generating Liapunov functions for autonomous and nonautonomous systems. Three examples are given. An explicit expression for V and V̇ for second-order systems is given in terms of the components of a system ẋ = f and any, arbitrary real function p(x ; t). A Liapunov function is generated for a more general class of third-order systems than any which has been given heretofore. Also, it is shown how the basic vector format may be applied to inverse Liapunov problems.