Demonstrate by the Lyapunov method

Question

Demonstrate by the Lyapunov method or any other method that a system described by the equation $ \ddot{y} + h (\dot{y}) + f (y) = 0 $ is globally asymptotically stable if the functions $ f, h $ satisfy $ yf (y) \geq \xi y^{2}$ for some $ \xi> 0 $ and $ \dot{y}h(\dot{y})> 0 $, $\forall\dot{y} \neq0 $

Answer 1

One obvious Lyapunov function is $$V=\frac12\dot y^2+F(y),$$ with $F$ an anti-derivative of $f$. Then $$\dot V=-h(\dot y)\dot y<0,$$ so that the level of $V$ continuously decreases along solutions. Now you only need to explore the shape of the level sets of the minima of $V$.

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Setting $F(0)=0$ we get $$F(y)=\int_0^1f(sy)y\,ds\ge\int_0^1 s\,ds\,ξy^2=\frac12ξy^2$$ which not only shows positivity for $y\ne 0$ but coercivity. The minimum of $V$ is then $0$ at $(y,\dot y)=(0,0)$ and all trajectories converge towards that point.