Stability of Equilibria in Modified-Gradient Systems

Motivated by questions in biology, we investigate the stability of equilibria of the dynamical system ${\mathbf{x}}^{\mathrm{\prime }}=P(t)\mathrm{\nabla }f(x)$ which arise as critical points of $f$, under the assumption that $P(t)$ is positive semi-definite. It is shown that the condition ${\int }^{\mathrm{\infty }}{\lambda }_1(P(t))dt=\mathrm{\infty }$, 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.