Some Remarks on Positive/Negative Feedback

In the context of linear control systems, a commonly-held intuition is that negative and positive feedback cannot both be stability enhancing. The canonical linear prototype is the scalar system $\dot{x}=u$ which, under negative linear feedback $u=-kx$ ($k>0$) is exponentially stable for all $k>0$, whereas the lack of exponential instability of the (marginally stable) uncontrolled system is amplified by positive feedback $u=kx$ ($k>0)$. By contrast, for nonlinear systems it is shown, by example, that this intuitive dichotomy may fail to hold.