# Mikhailov criterion

All roots of a polynomial

$$P(z)=z^n+a_{n-1}z^{n-1}+\dotsb+a_0$$

with real coefficients have strictly negative real part if and only if the complex-valued function $z=P(i\omega)$ of a real variable $\omega\in[0,\infty)$ describes a curve (the Mikhailov hodograph) in the complex $z$-plane which starts on the positive real semi-axis, does not hit the origin and successively generates an anti-clockwise motion through $n$ quadrants. (An equivalent condition is: The radius vector $P(i\omega)$, as $\omega$ increases from $0$ to $+\infty$, never vanishes and monotonically rotates in a positive direction through an angle $n\pi/2$.)

This criterion was first suggested by A.V. Mikhailov . It is equivalent to the Routh–Hurwitz criterion; however, it is geometric in character and does not require the verification of determinant inequalities (see , ). The Mikhailov criterion gives a necessary and sufficient condition for the asymptotic stability of a linear differential equation of order $n$,

\begin{equation*} x ^ { ( n ) } + a _ { n - 1} z ^ { ( n - 1 ) } + \dots + a _ { 0 } x = 0, \end{equation*}

with constant coefficients, or of a linear system

\begin{equation*} \dot { x } = A x , \quad x \in {\bf R} ^ { n }, \end{equation*}

with a constant matrix $A$, the characteristic polynomial of which is $P ( z )$ (see ).

Mikhailov's criterion is one of the frequency criteria for the stability of linear systems of automatic control (closely related to, for example, the Nyquist criterion). A generalization of Mikhailov's criterion is known for systems of automatic control with delay, for impulse systems (see ), and there is also an analogue of Mikhailov's criterion for non-linear control systems (see ).

How to Cite This Entry:
Mikhailov criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mikhailov_criterion&oldid=49921
This article was adapted from an original article by r equation','../w/w097310.htm','Whittaker equation','../w/w097840.htm','Wronskian','../w/w098180.htm')" style="background-color:yellow;">N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article