Consider a simple pendulum that has damping friction between the pivot point and the rod see

Figure $1.$ The dynamics parameters are: the mass of the ball is $($m$),$ the length of the rod is $($L$),$

the mass of the rod is neglected, and the damping coefficient of the friction is $($c$).$Figure $1$: a simple pendulum system.

a$.$ Find the nonlinear state$-$space representation.

b$.$ Study the Lyapunov stability for the simple pendulum system if the suggested

Lyapunov function is:

$V(z)=\frac{mgL}{J}(1-cos)+\frac{1}{2}{}^{}{?}^2.$