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 $L[x]=0$ .

Keywords

Lyapunov functions; Nonlinear differential equations; Asymptotic stability; Differential equations; Integral equations; Sufficient conditions;

fLanguage

English

Journal_Title

Automatic Control, IEEE Transactions on

Publisher

ieee

ISSN

0018-9286

Type

jour

DOI

10.1109/TAC.1965.1098187

Filename

1098187

Link To Document

https://search.isc.ac/dl/search/defaultta.aspx?DTC=49&DC=790038