On the stability of certain nonlinear differential equations

A technique is described, involving integration by parts, for constructing Liapunov functions for a class of third- and fourth-order differential equations. The procedure makes use of a number of "differential moments"
M_{n} = \int\min{t_{0}}\max{t} (d^{n}x/dt^{n})L[x]dt
, where
L[x] = 0
is the differential equation under consideration. The moment equations
M_{n}=0 (n=0, 1, 2, ...)
are manipulated to obtain a single expression
[V(x,\dot{x},\ddot{x},... )]\min{t_{0}}\max{t} = \int\min{t_{0}}\max{t} k(x,\dot{x},\ddot{x}...)dt
. Conditions are applied to the functions involved in
L[x]
so as to make
V
positive definite and
k
negative semidefinite, in which case
V(x,\dot{x},\ddot{x} ... )
will be a Liapunov function for
L[x]=0
.