Stable polynomial

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

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

• Necessary condition: a Hurwitz stable polynomial has coefficients of the same sign.
• Sufficient condition: a polynomial with coefficients such that:

• 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