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arXiv · 2510.24555

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

Dinesh Kumar Keshari·Shubhankar Mandal·Avijit Pal
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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.

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Dinesh Kumar Keshari, Shubhankar Mandal, Avijit Pal. 2025-10-28. Function Theory and necessary conditions for a Schwarz lemma related to $μ$-Synthesis Domains. https://arxiv.org/abs/2510.24555

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