Stability of Equilibria in Modified-Gradient Systems

TL;DR

This paper studies the stability of equilibria in modified-gradient dynamical systems influenced by biological motivations, establishing conditions for uniform asymptotic stability based on eigenvalue integrals of the system matrix.

Contribution

It introduces a new stability criterion for systems with time-varying positive semi-definite matrices, linking eigenvalue integrals to equilibrium stability.

Findings

The integral of the smallest eigenvalue over time determines stability.

Conditions for uniform asymptotic stability are established.

Provides insights into the attraction basins of equilibria.

Abstract

Motivated by questions in biology, we investigate the stability of equilibria of the dynamical system $\mathbf{x}^{\prime}=P(t)\nabla f(x)$ which arise as critical points of $f$, under the assumption that $P(t)$ is positive semi-definite. It is shown that the condition $\int^{\infty}\lambda_{1}(P(t))~dt=\infty$, where $\lambda_{1}(P(t))$ is the smallest eigenvalue of $P(t)$, plays a key role in guaranteeing uniform asymptotic stability and in providing information on the basis of attraction of those equilibria.