# Mikhailov criterion

All roots of a polynomial

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

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 [1]. It is equivalent to the Routh–Hurwitz criterion; however, it is geometric in character and does not require the verification of determinant inequalities (see [2], [3]). The Mikhailov criterion gives a necessary and sufficient condition for the asymptotic stability of a linear differential equation of order ,

with constant coefficients, or of a linear system

with a constant matrix , the characteristic polynomial of which is (see [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). A generalization of Mikhailov's criterion is known for systems of automatic control with delay, for impulse systems (see [5]), and there is also an analogue of Mikhailov's criterion for non-linear control systems (see [6]).

#### References

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