Lyapunov-like Conditions of Forward Invariance and Boundedness for a Class of Unstable Systems

TL;DR

This paper introduces Lyapunov-like conditions to determine boundedness and convergence of solutions in a class of unstable systems, applicable to nonlinear control and neural oscillator synchronization.

Contribution

It provides novel Lyapunov-like criteria for unstable systems that do not rely on input-output characterizations or convergence rate estimates.

Findings

Criteria for boundedness and convergence derived

Domains of initial conditions identified for solutions leaving neighborhoods

Applications demonstrated in neural oscillator phase synchronization

Abstract

We provide Lyapunov-like characterizations of boundedness and convergence of non-trivial solutions for a class of systems with unstable invariant sets. Examples of systems to which the results may apply include interconnections of stable subsystems with one-dimensional unstable dynamics or critically stable dynamics. Systems of this type arise in problems of nonlinear output regulation, parameter estimation and adaptive control. In addition to providing boundedness and convergence criteria the results allow to derive domains of initial conditions corresponding to solutions leaving a given neighborhood of the origin at least once. In contrast to other works addressing convergence issues in unstable systems, our results require neither input-output characterizations for the stable part nor estimates of convergence rates. The results are illustrated with examples, including the analysis of phase synchronization of neural oscillators with heterogenous coupling.