About root-clustering in sophisticated regions

This paper presents a new criteria for matrix root-clustering in sophisticated, multiply-connected and non-convex regions of the complex plane. A concept of the closed forbidden region is defined as a complement of the root-clustering region to C and the concepts of the forbidden subregions and the expanded root-clustering regions are introduced to formulate the main results of the paper. The set of the modified three-parametrical Cassini regions are offered to use as the subregions covering the forbidden region. In these terms the generalizations of the classical Gutman theorem and the Jury-Ahn theorem onto the intersection of the systems of nonlinear algebraic inequalities are obtained. An application to the problem of root-clustering in outside of a system of the forbidden frequency bands is shown for illustration. Key–Words: Matrix root clustering, Lyapunov inequality, Forbidden subregions, expanded root-clustering regions, Modified Cassini regions