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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2510.24555 |
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| _version_ | 1866917048169267200 |
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| author | Keshari, Dinesh Kumar Mandal, Shubhankar Pal, Avijit |
| author_facet | Keshari, Dinesh Kumar Mandal, Shubhankar Pal, Avijit |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_24555 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Function Theory and necessary conditions for a Schwarz lemma related to $μ$-Synthesis Domains Keshari, Dinesh Kumar Mandal, Shubhankar Pal, Avijit Functional Analysis 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. |
| title | Function Theory and necessary conditions for a Schwarz lemma related to $μ$-Synthesis Domains |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2510.24555 |