Method of forbidden regions in the dynamic system matrices root clustering problem

New criteria for root-clustering of the dynamic system matrices in multiply connected regions were presented. A concept of the forbidden region was introduced, and a generalization of the Lyapunov–Jury theorem onto the multiply connected regions was proposed. The necessary and sufficient conditions for root-clustering outside the forbidden region were established as a sequence of matrix inequalities. Construction of the conditions for root-clustering outside the frequency bands and robust root-clustering in multiply connected region was illustrated by way of examples.