Some results on existence and uniqueness of solutions of nonlinear networks

Author

Fujisawa, Toshio ; Kuh, Ernest S.

Volume

18

Issue

5

fYear

1971

fDate

9/1/1971 12:00:00 AM

Firstpage

501

Lastpage

506

Abstract

This paper deals with nonlinear networks which can be characterized by the equation $f(x) = y$ , where $f(\\cdot)$ maps the real Euclidean $n$ -space $R^{n}$ into itself and is assumed to be continuously differentiable $x$ is a point in $R^{n}$ and represents a set of chosen network variables, and $y$ is an arbitrary point in $R^{n}$ and represents the input to the network. The authors derive sufficient conditions for the existence of a unique solution of the equation for all $y \\in R^{n}$ in terms of the Jacobian matrix $\\partial f/ \\partial x$ . It is shown that if a set of cofactors of the Jacobian matrix satisfies a "ratio condition," the network has a unique solution. The class of matrices under consideration is a generalization of the class $P$ recently introduced by Fiedler and Pták, and it includes the familiar uniformly positive-definite matrix as a special case.

Keywords

Nonlinear network analysis & design; Nonlinear networks; Resistance networks; Circuit theory; Control engineering; Couplings; Jacobian matrices; Laboratories; Nonlinear equations; Resistors; Sufficient conditions; Tin;

fLanguage

English

Journal_Title

Circuit Theory, IEEE Transactions on

Publisher

ieee

ISSN

0018-9324

Type

jour

DOI

10.1109/TCT.1971.1083336

Filename

1083336