The Small-Gain Condition for Infinite Networks Modeled on $\ell^{\infty}$-Spaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911397951045632 |
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| author | Kawan, Christoph |
| author_facet | Kawan, Christoph |
| contents | In recent years, attempts have been made to extend nonlinear small-gain theorems for input-to-state stability (ISS) from finite networks to countably infinite networks with finite indegrees. Under specific assumptions about the interconnection gains and the ISS formulation, corresponding infinite-dimensional small-gain results have been proven. However, concerning these assumptions, the results are still too narrow to be considered a full extension of the state-of-the-art for finite networks. We take a step to closing this gap by developing a general technical framework within which the small-gain condition for both finite and infinite networks can be analyzed. This includes a thorough investigation of various monotone operators associated with a network and a specific ISS formulation. Our results extend and generalize the existing theory for finite networks, yield complete characterizations of the small-gain condition for specific ISS formulations, and show which obstacles still have to be overcome to obtain a complete theory for the most general infinite case. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_03925 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Small-Gain Condition for Infinite Networks Modeled on $\ell^{\infty}$-Spaces Kawan, Christoph Optimization and Control Dynamical Systems 93A15, 93B70, 93D09, 93D05, 47H05 In recent years, attempts have been made to extend nonlinear small-gain theorems for input-to-state stability (ISS) from finite networks to countably infinite networks with finite indegrees. Under specific assumptions about the interconnection gains and the ISS formulation, corresponding infinite-dimensional small-gain results have been proven. However, concerning these assumptions, the results are still too narrow to be considered a full extension of the state-of-the-art for finite networks. We take a step to closing this gap by developing a general technical framework within which the small-gain condition for both finite and infinite networks can be analyzed. This includes a thorough investigation of various monotone operators associated with a network and a specific ISS formulation. Our results extend and generalize the existing theory for finite networks, yield complete characterizations of the small-gain condition for specific ISS formulations, and show which obstacles still have to be overcome to obtain a complete theory for the most general infinite case. |
| title | The Small-Gain Condition for Infinite Networks Modeled on $\ell^{\infty}$-Spaces |
| topic | Optimization and Control Dynamical Systems 93A15, 93B70, 93D09, 93D05, 47H05 |
| url | https://arxiv.org/abs/2503.03925 |