View source for Mikhailov criterion

<!--This article has been texified automatically. Since there was no Nroff source code for this article, 
the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist
was used.
If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category.

Out of 11 formulas, 11 were replaced by TEX code.-->

{{TEX|semi-auto}}{{TEX|done}}
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 [[#References|[1]]]. It is equivalent to the [[Routh–Hurwitz criterion|Routh–Hurwitz criterion]]; however, it is geometric in character and does not require the verification of determinant inequalities (see [[#References|[2]]], [[#References|[3]]]). 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 [[#References|[4]]]).

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|Nyquist criterion]]). A generalization of Mikhailov's criterion is known for systems of automatic control with delay, for impulse systems (see [[#References|[5]]]), and there is also an analogue of Mikhailov's criterion for non-linear control systems (see [[#References|[6]]]).

====References====
<table><tr><td valign="top">[1]</td> <td valign="top">  A.V. Mikhailov,   ''Avtomat. i Telemekh.'' , '''3'''  (1938)  pp. 27–81</td></tr><tr><td valign="top">[2]</td> <td valign="top">  N.G. Chebotarev,   N.N. Meiman,   "The Routh–Hurwitz problem for polynomials and entire functions"  ''Trudy Mat. Inst. Steklov.'' , '''76'''  (1949)  (In Russian)</td></tr><tr><td valign="top">[3]</td> <td valign="top">  M.A. Lavrent'ev,   B.V. Shabat,   "Methoden der komplexen Funktionentheorie" , Deutsch. Verlag Wissenschaft.  (1967)  (Translated from Russian)</td></tr><tr><td valign="top">[4]</td> <td valign="top">  B.P. Demidovich,   "Lectures on the mathematical theory of stability" , Moscow  (1967)  (In Russian)</td></tr><tr><td valign="top">[5]</td> <td valign="top">  L.S. Gnoenskii,   G.A. Kamenskii,   L.E. El'sgol'ts,   "Mathematical foundations of the theory of control systems" , Moscow  (1969)  (In Russian)</td></tr><tr><td valign="top">[6]</td> <td valign="top">  A. Blaquiére,   "Mécanique non-lineaire" , Gauthier-Villars  (1960)</td></tr></table>



====Comments====
Recently a generalization of stability criteria with respect to roots of polynomials has been found. It is named after V.L. Kharitonov [[#References|[a1]]]. The generalization is that the coefficients $a_i$ in the polynomial $p ( z ) = z ^ { n } + a _ { n  - 1} z ^ { n - 1 } + \ldots + a _ { 0 }$ take values in given intervals $[ a _ { i } ^ { - } , a _ { i } ^ { + } ]$, $i = 0 , \ldots , n - 1$. The problem addressed by Kharitonov is whether all polynomials $p ( z )$ with $a _ { i } \in [ a _ { i } ^ { - } , a _ { i } ^ { + } ]$ are strictly stable. It turns out that the stability of only four specific polynomials has to be investigated in order to answer this question.

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  V.L. Kharitonov,   "Asymptotic stability of an equilibrium position of a family of systems of linear differential equations"  ''Differential Eq.'' , '''14''' :  11  (1978)  pp. 1483–1485  ''Differentsial'nye Uravnen.'' , '''14''' :  11  (1978)  pp. 2086–2088</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  B.R. Barmish,   "New tools for robustness analysis" , ''IEEE Proc. 27th Conf. Decision and Control, Austin, Texas, December 1988'' , IEEE  (1988)  pp. 1–6</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  S. LaSalle,   "Stability by Liapunov's direct method" , Acad. Press  (1961)</td></tr></table>