I'm trying to solve question 6 in this past exam. The setting is the Liénard ODE $$u''+c(u)u'+g(u)=0 $$ with $c(u) \geq 0$, $g(0)=0$ and $ug(u)>0$ in some punctured neighborhood of $u=0$.
Part (a) asks to show that the origin $(u,u')=(0,0)$ is (Lyapunov) stable. Part (b) asks to show that in the special case where $c(u) \equiv 1$, the origin is asymptotically stable.
My attempt:
For part (a), consider the function (taken from Khalil's Nonlinear systems, p. 184) $$V(u,u'):=\int_0^{u} g(v) \mathrm{d} v+\frac{1}{2} u'^2 $$ in a domain where $|u|<\delta$ is small enough so that $ug(u)>0$ for $0 \neq |u|<\delta$. We have $$\frac{\mathrm{d}}{\mathrm{d} t} V(u(t),u'(t)) =g(u)u'+u' u''=u'(-c(u)u')=-c(u) u'^2 \leq 0. $$ Hence $V$ is a Lyapunov function, and solutions remain in the interior of the contour lines of $V$ if they start with sufficiently small $|u|$.
For part (b), using the same Lyapunov function we get $$\frac{\mathrm{d}}{\mathrm{d}t} V(u(t),u'(t))=-u'^2 \leq 0 $$ with equality happening if and only if $u'=0$. However, at such points we see that $$u''=-g(u) $$ so that the first derivative of $u$ changes sign every time it becomes zero. Corollary 4.1 in Khalil's text then gives that the origin is asymptotically stable.
I'd like to know whether my solution is correct. Any other simple solutions are welcome as well. Thank you!
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.