A. P. Zhabko and V. L. Kharitonov, "Necessary and sufficient conditions for the stability of a linear family of polynomials", Automation and Remote Control, 1994, Vol. 55, No. 10, pp. 1496-1503.

Stability of a linear system $\dot x=Ax$ is equivalent to the fact that all roots of the characteristic polynomial $p(\lambda)=\det(A-\lambda I)$ have negative real parts. This property of roots is easily checkable. (Because of this equivalence, a polynomial $p(\lambda)=a_0\lambda^n+a_1\lambda^{n-1}+\ldots+a_n$ with this property is called stable.)

For real-life systems, we often do not know the exact values of the coefficients of the matrix $A$ and therefore, we do not know the exact coefficients $a_i$ of the characteristic polynomial $p(\lambda)$. How can we then check whether the system is stable? It is known that if the set $S$ of possible values of the coefficients $\vec a=(a_0,a_1,\ldots,a_n)$ is a convex polytope, then the stability for all $\vec a\in S$ is equivalent to the stability for all $\vec a$ from all edges (one-dimensional faces) of the polytope $S$. In the paper under review, a new method is described for checking whether all polynomials from a given edge are stable.