مرکز منطقه ای اطلاع رساني علوم و فناوري - On the stability of certain nonlinear differential equations

DocumentCode :

790038

Title :

On the stability of certain nonlinear differential equations

Author :

Ponzo, Peter J.

Author_Institution :

University of Waterloo, Waterloo, Ontario, Canada

Volume :

Issue :

fYear :

1965

fDate :

10/1/1965 12:00:00 AM

Firstpage :

470

Lastpage :

472

Abstract :

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 $L[x]=0$ .