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

TL;DR

This paper characterizes specific complex domains related to structured singular values in control theory, analyzing their geometric properties and establishing conditions for a Schwarz lemma applicable to these domains.

Contribution

It provides a detailed geometric and analytic characterization of the domains associated with $ta$-synthesis, including convexity properties and necessary conditions for Schwarz lemmas.

Findings

Domains are simply connected but not convex or circular.

Established polynomial and linear convexity of the closure domains.

Derived necessary conditions for Schwarz lemmas on these 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.