On the Generalized Stability of the Convex Hull of Two Matrices.

Abstract

Contingent on the existence of certain affine maps, necessary and sufficient conditions for the eigenvalues of all matrices belonging to the convex hull of two given matrices to lie in a subset of the complex plane are obtained. Such maps are identified for every balanced convex domain D with polygonal boundary and conclusive D-stability criteria are obtained. In the case when D is the open left half plane, the computational complexity of the new test is somewhat lower than that of a previously proposed criterion. For nonpolygonal balanced convex domains, conditions that are as close to being necessary and sufficient as desired may be obtained via a suitable approximation of these domains by polygonal ones.