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: 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 stabilityof 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 theoryof differential and difference equations. A linear, time-invariant system (see LTI system theory) 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. obtained after the Möbius transformation z ↦ z + 1 z − 1 {displaystyle zmapsto {{z+1} over {z-1}}} which maps the left half-plane to the open unit disc: P is Schur stable if and only if Q is Hurwitz stable and P ( 1 ) ≠ 0 {displaystyle P(1) eq 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, the Jury test or the Bistritz test.