In the context of thecharacteristic polynomial of adifferential equation ordifference equation, apolynomial is said to bestable if either:

• all its roots lie in theopen lefthalf-plane, or
• all its roots lie in theopenunit disk.

The first condition providesstability forcontinuous-time linear systems, and the second case relates to stabilityofdiscrete-time linear systems. A polynomial with the first property is called at times aHurwitz-stable polynomial and with the second property a Schur-stable polynomial. Stable polynomials arise incontrol theory and in mathematical theoryof differential and difference equations. A linear,time-invariant system (seeLTI system theory) is said to beBIBO stable if every bounded input produces bounded output. A linear system is BIBO stable if its characteristic polynomial is stable. The denominator is required to be Hurwitz stable if the system is in continuous-time and Schur stable if it is in discrete-time. In practice, stability is determined by applying any one of severalstability criteria.

• TheRouth–Hurwitz theorem provides an algorithm for determining if a given polynomial is Hurwitz stable, which is implemented in theRouth–Hurwitz andLiénard–Chipart tests.
• To test if a given polynomialP (ofdegreed) is Schur stable, it suffices to apply this theorem to the transformed polynomial

${\displaystyle Q(z)=(z-1{)}^{d}P\left(\frac{z+1}{z-1}\right)}$

obtained after theMöbius transformation${\displaystyle z\mapsto \frac{z+1}{z-1}}$ which maps the left half-plane to the open unit disc:P is Schur stable if and only ifQ is Hurwitz stable and${\displaystyle P(1)\ne 0}$. For higher degree polynomials the extra computation involved in this mapping can be avoided by testing the Schur stability by the Schur-Cohn test, theJury test or theBistritz test.

• Necessary condition: a Hurwitz stable polynomial (withrealcoefficients) has coefficients of the same sign (either all positive or all negative).
• Sufficient condition: a polynomial${\displaystyle f(z)=a_0+a_{1}z+\cdots +a_n{z}^n}$ with (real) coefficients such that

${\displaystyle a_n>a_{n-1}>\cdots >a_0>0,}$

is Schur stable.

• Product rule: Two polynomialsf andg are stable (of the same type)if and only if the productfg is stable.
• Hadamard product: The Hadamard (coefficient-wise) product of two Hurwitz stable polynomials is again Hurwitz stable.^[1]

• ${\displaystyle 4z^3+3z^2+2z+1}$ is Schur stable because it satisfies the sufficient condition;
• ${\displaystyle z^{10}}$ is Schur stable (because all its roots equal 0) but it does not satisfy the sufficient condition;
• ${\displaystyle z^2-z-2}$ is not Hurwitz stable (its roots are −1 and 2) because it violates the necessary condition;
• ${\displaystyle z^2+3z+2}$ is Hurwitz stable (its roots are −1 and −2).
• The polynomial${\displaystyle z^4+z^3+z^2+z+1}$ (with positive coefficients) is neither Hurwitz stable nor Schur stable. Its roots are the four primitive fifthroots of unity

${\displaystyle z_k=\cos \left(\frac{2\pi k}{5}\right)+i\sin \left(\frac{2\pi k}{5}\right),k=1,\dots ,4.}$

Note here that

${\displaystyle \cos (2\pi /5)=\frac{\sqrt{5}-1}{4}>0.}$

It is a "boundary case" for Schur stability because its roots lie on theunit circle. The example also shows that the necessary (positivity) conditions stated above for Hurwitz stability are not sufficient.

Just as stable polynomials are crucial for assessing the stability of systems described by polynomials, stability matrices play a vital role in evaluating the stability ofsystems represented by matrices.

Asquare matrixA is called aHurwitz matrix if everyeigenvalue ofA has strictly negativereal part.

Schur matrices is an analogue of the Hurwitz matrices for discrete-time systems. A matrixA is a Schur (stable) matrix if its eigenvalues are located in theopen unit disk in thecomplex plane.

• Kharitonov region
• Stability criterion
• Stability radius

• Mathworld page

• Stability theory
• Polynomials

• Articles with short description
• Short description matches Wikidata