A generalized scheme for constructing Lyapunov functions from first integrals

A heuristic scheme is described for constructing Lyapunov v-functions, gene ralizing the classical method for constructing these functions from the fir st integrals of the equations of motion under investigation (or from the in tegrals of a comparison system). It is shown that the generalized scheme in herits a characteristic feature of the classical method: the Lyapunov funct ions are constructed as solutions of a certain completely integrable partia l differential equation (or system of such equations). The form of this equ ation and its order are uniquely defined by a non-degenerate multi-paramete r function V(x, alpha) + alpha (q), x is an element of R-n, alpha is an ele ment of Rq-1 (where alpha is a parameter vector), which generalizes the cla ssical linear combination of integrals. Methods are described for represent ing v-functions, in the course of which the traditional methods (the method of Chetayev combinations of integrals and the construction of Lyapunov fun ctions as a non-linear function of integrals) are augmented by geometrical constructions in which the v-functions are sought in the form of envelopes of certain subfamilies of the function V(x, alpha) + alpha (q). The general ized scheme serves as a basis for deriving new, simple criteria for the asy mptotic stability of the trivial solution in a transcendental problem of th e stability of a system with two degrees of freedom in the critical case of two pairs of pure imaginary roots at 1 : 1 resonance (the case of simple e lementary divisors). (C) 2001 Elsevier Science Ltd. All rights reserved.