Further Results on Lyapunov-Like Conditions of Forward Invariance and Boundedness for a Class of Unstable Systems

TL;DR

This paper introduces algebraic conditions for local convergence to unstable equilibria in nonlinear systems, extends Lyapunov's method for systems with zero eigenvalues, and shows persistent excitation ensures exponential convergence to the origin.

Contribution

It provides new algebraic criteria for convergence to unstable equilibria, extends Lyapunov's method, and links persistent excitation to exponential convergence in certain nonlinear systems.

Findings

Algebraic conditions for local convergence to unstable equilibria.

Persistent excitation guarantees exponential convergence to the origin.

Conditions from prior work are shown to be tight.

Abstract

We provide several characterizations of convergence to unstable equilibria in nonlinear systems. Our current contribution is three-fold. First we present simple algebraic conditions for establishing local convergence of non-trivial solutions of nonlinear systems to unstable equilibria. The conditions are based on the earlier work (A.N. Gorban, I.Yu. Tyukin, E. Steur, and H. Nijmeijer, SIAM Journal on Control and Optimization, Vol. 51, No. 3, 2013) and can be viewed as an extension of the Lyapunov's first method in that they apply to systems in which the corresponding Jacobian has one zero eigenvalue. Second, we show that for a relevant subclass of systems, persistency of excitation of a function of time in the right-hand side of the equations governing dynamics of the system ensure existence of an attractor basin such that solutions passing through this basin in forward time converge to the origin exponentially. Finally we demonstrate that conditions developed in (A.N. Gorban, I.Yu. Tyukin, E. Steur, and H. Nijmeijer, SIAM Journal on Control and Optimization, Vol. 51, No. 3, 2013) may be remarkably tight.