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" <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Mₙ = ∫min{t₀}max{t} (dⁿx/dtⁿ)L[x]dt</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L[x] = 0</tex> is the differential equation under consideration. The moment equations <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Mₙ=0 (n=0, 1, 2, ...)</tex> are manipulated to obtain a single expression <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[V(x,ẋ,ẍ,... )]min{t₀}max{t} = ∫min{t₀}max{t} k(x,ẋ,ẍ...)dt</tex> . Conditions are applied to the functions involved in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L[x]</tex> so as to make <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">V</tex> positive definite and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> negative semidefinite, in which case <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">V(x,ẋ,ẍ ... )</tex> will be a Liapunov function for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L[x]=0</tex> .