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A necessary and sufficient condition for the real parts of all roots of an equation

$$ \tag{* }
\lambda  ^ {n} + a _ {1} \lambda ^ {n - 1 } + \dots + a _ {n}  =  0
$$

to be negative.

A stability criterion is used in applying Lyapunov's theorem on the stability of the first approximation to a fixed point of an autonomous system of differential equations (cf. [[Lyapunov stability|Lyapunov stability]]). The most commonly used stability criterion is the Routh–Hurwitz criterion or Hurwitz criterion: For the real parts of all roots of the equation (*) to be negative it is necessary and sufficient that the inequalities  $  \Delta _ {i} > 0 $, 
$  i \in \{ 1 \dots n \} $, 
be satisfied, where

$$ 
\Delta _ {1} = a _ {1} ,\ \ 
\Delta _ {2} = \left |

\begin{array}{ll}
a _ {1}  & 1  \\
a _ {3}  &a _ {2}  \\
\end{array}
 \ 
\right | ,\ \ 
\Delta _ {3} = \left | 
\begin{array}{lll}
a _ {1}  & 1  & 0  \\
a _ {3}  &a _ {2}  &a _ {1}  \\
a _ {5}  &a _ {4}  &a _ {3}  \\
\end{array}
 \ 
\right | \dots
$$

are the principal diagonal minors of the matrix

$$ 
\left \| 
\begin{array}{lllllllll}
a _ {1}  & 1  & 0  & 0  & 0  & 0  &\cdot  &\cdot  & 0  \\
a _ {3}  &a _ {2}  &a _ {1}  & 1  & 0  & 0  &\cdot  &\cdot  & 0  \\
a _ {5}  &a _ {4}  &a _ {3}  &a _ {2}  &a _ {1}  & 1  &\cdot  &\cdot  & 0  \\
\cdot  &\cdot  &\cdot  &\cdot  &\cdot  &\cdot  &\cdot  &\cdot  &\cdot  \\
 0  & 0  & 0  & 0  & 0  & 0  &\cdot  &\cdot  &a _ {n}  \\
\end{array}
 \ 
\right \|
$$

(on the main diagonal of this matrix there stand  $  a _ {1} \dots a _ {n} $; 
for  $  i > n $, 
$  a _ {i} = 0 $).

For  $  n = 2 $
the Routh–Hurwitz stability criterion takes a particularly simple form: For the real parts of the roots of  $  \lambda  ^ {2} + a _ {1} \lambda + a _ {2} = 0 $
to be negative it is necessary and sufficient that the coefficients of the equation be positive:  $  a _ {1} > 0 $, 
$  a _ {2} > 0 $.

For each  $  n \in \mathbf N $, 
for the real parts of all roots of the equation (*) to be negative it is necessary (but for  $  n > 2 $
not sufficient) that all coefficients of the equation be positive:  $  a _ {i} > 0 $, 
$  i \in \{ 1 \dots n \} $. 
If at least one of the determinants  $  \Delta _ {i} $, 
$  i \in \{ 1 \dots n \} $, 
is negative, then there is a root of (*) with positive real part (this assertion is used in applying Lyapunov's theorem on the instability of the first approximation to a fixed point of an autonomous system of differential equations, cf. [[Lyapunov stability|Lyapunov stability]]). If  $  \Delta _ {i} \geq  0 $
for all  $  i \in \{ 1 \dots n \} $, 
but  $  \Delta _ {i} = 0 $
for a certain  $  i \in \{ 1 \dots n \} $, 
then the location of the roots of the equation (*) relative to the imaginary axis can also be described without finding the roots (cf. [[#References|[5]]], [[#References|[8]]], Chapt. XVI, Sect. 8).

Much simpler in applications is the Liénard–Chipart criterion: For the real parts of all roots of the equation (*) to be negative it is necessary and sufficient that the following inequalities hold:  $  a _ {i} > 0 $, 
$  i \in \{ 1 \dots n \} $, 
$  \Delta _ {n - 2i + 1 }  > 0 $, 
$  i \in \{ 1 \dots [ n/2] \} $(
the determinant  $  \Delta _ {i} $
is the same as in the Routh–Hurwitz criterion).

Hermite's criterion (historically the first, cf. [[#References|[1]]], [[#References|[10]]], Sect. 3.1) allows one to determine with the help of a finite number of arithmetic operations on the coefficients of (*) whether all roots of this equation have negative real parts. The Routh–Hurwitz criterion formulated above is a modification of Hermite's criterion found by A. Hurwitz. A Lyapunov stability criterion is also known (cf. [[#References|[3]]], [[#References|[8]]], Chapt. XVI, Sect. 5, [[#References|[10]]], Sect. 3.5).

For a study of the stability of fixed points of differentiable mappings (autonomous systems with discrete time) as well as for a study of [[Orbit stability|orbit stability]] of closed trajectories of autonomous systems of differential equations one has to apply necessary and sufficient conditions for the absolute values of all roots of the equation (*) to be less than one. This criterion is obtained from the above-mentioned stability criterion by the mapping  $  \lambda \mapsto ( \lambda + 1)/( \lambda - 1) $
from the open unit disc onto the open left half-plane (cf. [[#References|[10]]], Sect. 3.2).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Hermite,   "Sur le nombre des racines d'une équation algébrique comprise entre des limites donnés"  ''J. Reine Angew. Math.'' , '''52'''  (1856)  pp. 39–51</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.J. Routh,   "A treatise on the stability of a given state of motion" , Macmillan  (1877)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.M. Lyapunov,   "Stability of motion" , Acad. Press  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A. Hurwitz,   "Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt"  ''Math. Ann.'' , '''46'''  (1895)  pp. 273–284</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  L. Orlando,   "Sul problema di Hurwitz relative alle parti reali delle radici di un'equazione algebrica"  ''Math. Ann.'' , '''71'''  (1911)  pp. 233–245</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A. Liénard,   M.H. Chipart,   "Sur le signe de la partie réelle des racines d'une équation algébrique"  ''J. Math. Pure Appl. (6)'' , '''10'''  (1914)  pp. 291–346</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  N.G. Chetaev,   "Stability of motion" , Moscow  (1965)  (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  F.R. [F.R. Gantmakher] Gantmacher,   "The theory of matrices" , '''1''' , Chelsea, reprint  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  B.P. Demidovich,   "Lectures on the mathematical theory of stability" , Moscow  (1967)  (In Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  E. Jury,   "Inners and stability of dynamic systems" , Wiley  (1974)</TD></TR></table>

====Comments====
See also [[Mikhailov criterion|Mikhailov criterion]], which is equivalent to the Routh–Hurwitz criterion, but formulated in terms of the curve obtained from (*) by letting  $  \lambda $
vary over the positive imaginary axis.

In control theory (robust control) one is often concerned with the stability of a whole family of polynomials rather than a single one. Stability results pertaining to this situation are generally known as Kharitonov-type theorems.

The original Kharitonov theorem, [[#References|[a1]]], [[#References|[a2]]], can be stated as follows. Let  $  P( s;  q) $
be the family of polynomials

$$ 
P( s;  q)  =  q _ {0} + q _ {1} s + \dots + q _ {n} s  ^ {n} ,
$$

where each  $  q _ {i} $
ranges over a given closed interval  $  [ q _ {i}  ^ {-} , q _ {i}  ^ {+} ] $. 
Form the four polynomials

$$ 
K _ {1} ( s)  = \ 
q _ {0}  ^ {-} + q _ {1}  ^ {-} s +
q _ {2}  ^ {+} s  ^ {2} +
$$

$$ 
+ 
q _ {3}  ^ {+} s  ^ {3} + q _ {4}  ^ {-} s  ^ {4} + q _ {5}  ^ {-} s  ^ {5} + q _ {6}  ^ {+} s  ^ {6} + \dots ,
$$

$$ 
K _ {2} ( s)  =  q _ {0}  ^ {+} + q _ {1}  ^ {+} s + q _ {2}  ^ {-} s  ^ {2} +
$$

$$ 
+ 
q _ {3}  ^ {-} s  ^ {3} + q _ {4}  ^ {+} s  ^ {4} + q _ {5}  ^ {+} s  ^ {5} + q _ {6}  ^ {-} s  ^ {6} + \dots ,
$$

$$ 
K _ {3} ( s)  =  q _ {0}  ^ {+} + q _ {1}  ^ {-} s + q _ {2}  ^ {-} s  ^ {2} +
$$

$$ 
+ 
q _ {3}  ^ {+} s  ^ {3} + q _ {4}  ^ {+} s  ^ {4} + q _ {5}  ^ {-} s  ^ {5} + q _ {6}  ^ {-} s  ^ {6} + \dots ,
$$

$$ 
K _ {4} ( s)  =  q _ {0}  ^ {-} + q _ {1}  ^ {+} s + q _ {2}  ^ {+} s  ^ {2} +
$$

$$ 
+ 
q _ {3}  ^ {-} s  ^ {3} + q _ {4}  ^ {-} s  ^ {4} + q _ {5}  ^ {+} s  ^ {5} + q _ {6}  ^ {+} s  ^ {6} + \dots .
$$

Then every polynomial  $  P( s;  q) $, 
$  q _ {i}  ^ {-} \leq  q _ {i} \leq  q _ {i}  ^ {+} $, 
has its zeros strictly in the left half-plane if and only if the four polynomials  $  K _ {i} ( s) $, 
$  i = 1 \dots 4 $, 
have this property.

There is a large variety of similar theorems applying to other regions of allowed zeros, otherwise shaped families of polynomials (than cubes such as above), and discrete-time stability. Cf. [[#References|[a3]]] for a survey.

====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"  ''Diff. Uravn.'' , '''14''' :  11  (1978)  pp. 1483–1485  (In Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  V.L. Kharitonov,   "On a generalization of a stability criterion"  ''Akad. Nauk KazakhsSSR, Fiz.-Mat.'' , '''1'''  (1978)  pp. 53–57  (In Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  B.R. Barmish,   B. Ross,   "New tools for robustness analysis" , ''Proc. 27-th IEEE CDC'' , IEEE  (1988)  pp. 1–6</TD></TR></table>