Stability of Equilibria in Modified-Gradient Systems

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.