修正梯度系统平衡点的稳定性
Stability of Equilibria in Modified-Gradient Systems
摘要：
受生物学问题的启发，在假设$P(t)$是半正定的条件下，研究了动力系统$\mathbf{x}^{\prime}=P(t)\nabla f(x)$的平衡点的稳定性，这些平衡点是$\mathbf{x}^{\prime}=P(t)\nabla f(x)$的临界点.结果表明，条件$\int^{\infty}λ_{1}(P(t))~dt=\infty$，其中$λ_{1}(P(t))$是$P(t)$的最小特征值，起着关键作用，在保证一致渐近稳定性，并在提供信息的基础上，这些平衡。
英文摘要：
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 $\int^{\infty}λ_{1}(P(t))~dt=\infty$, where $λ_{1}(P(t))$ 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.
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