Function Theory and necessary conditions for a Schwarz lemma related to $\mu$-Synthesis Domains

Abstract

A subset of ${\mathbb{C}}^7$ (respectively, of ${\mathbb{C}}^5$) associated with the structured singular value ${\mu }_E$, defined on $3\times 3$ matrices, is denoted by $G_{E(3;3;1,1,1)}$ (respectively, by $G_{E(3;2;1,2)}$). In control engineering, the structured singular value ${\mu }_E$ plays a crucial role in analyzing the robustness and performance of linear feedback systems. We characterize the domain $G_{E(3;3;1,1,1)}$ and its closure ${\Gamma }_{E(3;3;1,1,1)}$, and employ realization formulas to describe both. The domain $G_{E(3;3;1,1,1)}$ and its closure are neither circular nor convex; however, they are simply connected. We provide an alternative proof of the polynomial and linear convexity of ${\Gamma }_{E(3;3;1,1,1)}$. Furthermore, we establish necessary conditions for a Schwarz lemma on the domains $G_{E(3;3;1,1,1)}$ and $G_{E(3;2;1,2)}$, and describe the relationships between these two domains as well as between their closed boundaries.