Exact determination of a Lyapunov function and the asymptotic stability domain

Abstract

The problem of the necessary and sufficient conditions for an exact direct determination of both a Lyapunov function υ and the asymptotic stability domain D of the zero state is solved for time-invariant non-linear continuous-time systems with continuous unique motions. The conditions provide a complete algorithm for their simultaneous determination. They permit arbitrary selection of a positive definite function p for which D +υ = −p has a solution υ in order to generate a system Lyapunov function υ from D +υ = −p as for linear time-invariant systems. This expresses their substantial difference from Lyapunov's original results and their developments so far. The latter are expressed in terms of the existence of a system Lyapunov function but the former are not. The theorems of the paper establish a new insight into, and a methodology for, a study of Lyapunov stability properties and other qualitative features of nonlinear dynamic systems. Simple examples illustrate the theorems and the procedure for their...