Routh-Hurwitz stability criterion

The Routh-Hurwitz criterion tests whether all roots of a real polynomial have negative real part, by sign conditions on determinants formed from its coefficients. Applied to a characteristic polynomial, it gives an algebraic test for strict linear asymptotic stability.

Consider the real family $s^4+2\gamma s^3+(K+{\gamma }^2)s^2+2L\gamma s+P{\gamma }^2$ with fixed $K,L,P$. The Routh-Hurwitz criterion reduces its nontrivial determinant to $4{\gamma }^2[L(K-L)+{\gamma }^2(L-P)]$. Strict decay for every finite $\gamma >0$ is equivalent to the displayed inequalities: the constant and slope must be nonnegative, with at least one positive, and coefficient positivity requires $P>0$. The stricter chain $K>L>P>0$ is sufficient but excludes valid equality cases. Neither equality case guarantees a uniform decay margin as $\gamma$ tends to zero or infinity. At $\gamma =0$ there is no asymptotic attraction.

A real $2\times 2$ Jacobian matrix has both eigenvalues in the open left half-plane exactly when its trace is negative and its determinant is positive. At zero trace the test no longer determines nonlinear stability.