Weakening of the sign-definiteness condition for the derivative in some theorems of Lyapunov's second method

Problems relating to the analysis of instability and asymptotic stability are considered for non-steady systems of ordinary differential equations, solved for the derivative. It is assumed that the right-hand sides of the system converge uniformly as the time increases without limit, tending to certain functions of the phase variables. Propositions are proved analogous to those of Lyapunov's second method [1-7] for steady systems, but the condition that the derivative of the Lyapunov function be sign-definite is relaxed. Instead, the derivative is required to be of constant sign, and a certain algebraic condition, which may always be verified directly, is imposed on the Lyapunov function. (C) 2004 Elsevier Ltd. All rights reserved.