Stable polynomial In the context of the characteristic polynomial of a differential equation or difference equation , a polynomial is said to be stable if either: The first condition provides stability for continuous-time linear systems , and the second case relates to stability of discrete-time linear systems. A polynomial with the first property is called at times a Hurwitz polynomial and with the second property a Schur polynomial . Stable polynomials arise in control theory and in mathematical theory of differential and difference equations . A linear, time-invariant system is said to be BIBO 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 several stability criteria .Properties The Routh–Hurwitz theorem provides an algorithm for determining if a given polynomial is Hurwitz stable, which is implemented in the Routh–Hurwitz and Liénard–Chipart tests. To test if a given polynomial P is Schur stable, it suffices to apply this theorem to the transformed polynomial obtained after the Möbius transformation which maps the left half-plane to the open unit disc: P is Schur stable if and only if Q is Hurwitz stable and. For higher degree polynomials the extra computation involved in this mapping can be avoided by testing the Schur stability by the Schur-Cohn test, the Jury test or the Bistritz test . Necessary condition: a Hurwitz stable polynomial has coefficients of the same sign . Sufficient condition: a polynomial with coefficients such that: is Schur stable. Product rule: Two polynomials f and g are stable if and only if the product fg is stable. Hadamard product: The Hadamard product of two Hurwitz stable polynomials is again Hurwitz stable.Examples is Schur stable because it satisfies the sufficient condition ; is Schur stable but it does not satisfy the sufficient condition; is not Hurwitz stable because it violates the necessary condition ; is Hurwitz stable. The polynomial is neither Hurwitz stable nor Schur stable. Its roots are the four primitive fifth roots of unity
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