Stable Polynomial

Stable Polynomial

A polynomial is said to be stable if either:

• all its roots lie in the open left half-plane, or
• all its roots lie in the open unit disk.

The first condition defines Hurwitz (or continuous-time) stability and the second one Schur (or discrete-time) stability. Stable polynomials arise in various mathematical fields, for example in control theory and differential equations. Indeed, a linear, time-invariant system (see LTI system theory) is said to be BIBO stable if and only if bounded inputs produce bounded outputs; this is equivalent to requiring that the denominator of its transfer function (which can be proven to be rational) 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. Stable polynomials are sometimes called Hurwitz polynomials and Schur polynomials. The latter should not be confused with Schur polynomial, which are multivariate polynomials, for which stability (in this context) is undefined.