Stability in a Liénard system

Consider the Lyapunov $$V=\frac{1}{2}\left(u'+\int_0^uc(v)dv\right)^2+\frac{1}{2}u'^2+2\int_0^ug(v)dv$$ with derivative $$\frac{d}{dt}V(u(t),u'(t))=-g(u)\left(u'+\int_0^uc(v)dv\right)-u'\left(c(u)u'+g(u))+2g(u\right)u'\\ =-g(u)u\int_0^1c(\theta u)d\theta-c(u)u'^2\leq 0$$ For $c(u)=1$ this yields $$\frac{d}{dt}V(u(t),u'(t))\leq -g(u)u-c(u)u'^2$$ which is negative definite and directly implies the asymptotic stability result.

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