proving the stability of the equilibrium point with a Lyapunov function

Question

The below nonlinear system is considered as a pendulum with a nonlinear damping coefficient: $$ \ddot y+(a+b\cos(y))\dot y+c\sin(y)=0, \qquad a\geq b\geq 0 $$ Use the energy of the whole system as a Lyapunov function to check the stability of the equilibrium point.

Answer 1

You have the actuated system $$ \ddot y+c\sin(y)=-(a+b\cos(y))\dot y $$ The sum of the kinetic and potential energy is given by $$V(y,\dot y)={\dot y}^2/2+\int c\sin y\,\mathrm{d}y.$$ The derivative of $V(y,\dot y)$ is $$\dot V(y,\dot y)=\dot y \ddot y+\dot yc\sin y=-\dot y c\sin y-{\dot y}^2(a+b\cos y)+\dot y c\sin y=-{\dot y}^2(a+b\cos y),$$ by substituting $\ddot y=-(a+b\cos(y))\dot y-c\sin(y)$. From $a\geq b\geq 0$ and $\cos y\geq -1$ follows $$\dot V(y,\dot y)\leq 0.$$