Function Theory and necessary conditions for a Schwarz lemma related to $μ$-Synthesis Domains
Dinesh Kumar Keshari, Shubhankar Mandal, Avijit Pal
TL;DR
The paper studies μ_E-synthesis domains associated with diagonal block-structured matrices, focusing on $G_{E(3;3;1,1,1)}\subset\mathbb{C}^7$ and $G_{E(3;2;1,2)}\subset\mathbb{C}^5$, and their closures $\Gamma_{E(3;3;1,1,1)}$ and $\Gamma_{E(3;2;1,2)}$. It establishes three parallel characterizations of $G_{E(3;3;1,1,1)}$—polynomial zero-set, rational-function, and realization-formula—along with corresponding results for the closures, and relates these domains to their $3\times3$-block tetrablock analogs. The authors prove non-convex, non-circular geometry with simple connectivity, and show polynomial and linear convexity for the closure $\Gamma_{E(3;3;1,1,1)}$ while describing boundary structures and analytic retracts connecting the $(3,3)$ and $(3,2)$ models. They derive Schwarz lemma-type necessary conditions for maps into $\Gamma_{E(3;3;1,1,1)}$ and $\Gamma_{E(3;2;1,2)}$, using one-variable reductions to the $G_{E(2;2;1,1)}$ (tetrablock) and Schur-Agler frameworks. The results deepen the understanding of μ_E-synthesis domains, providing concrete algebraic, analytic, and geometric characterizations that can inform robust control designs and complex-geometry studies in several variables.
Abstract
A subset of $\mathbb{C}^7$ (respectively, of $\mathbb{C}^5$) associated with the structured singular value $μ_E$, 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 $μ_E$ 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 $Γ_{E(3;3;1,1,1)}$, 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 $Γ_{E(3;3;1,1,1)}$. 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.