Stability theory of differential equations

Abstract

The problem of determining the stability of a set of linear differential equations has been of interest to mathematicians and engineers for a considerable length of time.

The problem is attacked by obtaining the characteristic equation of the original set of equations and determining the stability of this equation.

The stability of the characteristic equation is first considered in terms of a continued fraction expansion. Necessary and sufficient conditions are given for the characteristic equation to be stable.

The stability of the equation is then determined by means of a determinant sequence, which was the manner originally presented by A. Hurwitz in 1895.

The Nyquist criterion, which is a graphical method for determining whether the equation is stable, is then presented.

An example is given for each of the above methods to illustrate the procedure used in determining whether the equation is stable or unstable. Also included is a brief analysis of stability for non-linear equations.