(a) A special case of the Lienard equation of Problem 8 is
$$
\frac{d^{2} u}{d t^{2}}+\frac{d u}{d t}+g(u)=0
$$
where \(g\) satisfies the conditions of Problem 6 . Letting \(x=u, y=d u / d t,\)
show that the origin is a critical point of the resulting system. This
equation can be interpreted as describing the motion of a spring-mass system
with damping proportional to the velocity and a nonlinear restoring force.
Using the Liapunov function of Problem \(6,\) show that the origin is a stable
critical point, but note that even with damping we cannot conclude asymptotic
stability using this Liapunov function.
(b) Asymptotic stability of the critical point \((0,0)\) can be shown by
constructing a better Liapunov function as was done in part (d) of Problem 7 .
However, the analysis for a general function \(g\) is somewhat sophisticated and
we only mention that appropriate form for \(V\) is
$$
V(x, y)=\frac{1}{2} y^{2}+A y g(x)+\int_{0}^{x} g(s) d s
$$
where \(A\) is a positive constant to be chosen so that \(V\) is positive definite
and \(\hat{V}\) is negative definite. For the pendulum problem \([g(x)=\sin x]\)
use \(V\) as given by the preceding equation with \(A=\frac{1}{2}\) to show that
the origin is asymptotically stable. Hint: Use \(\sin x=x-\alpha x^{3} / 3 !\)
and \(\cos x=1-\beta x^{2} / 2 !\) where \(\alpha\) and \(\beta\) depend on \(x,\) but
\(0<\alpha<1\) and \(0<\beta<1\) for \(-\pi / 2