Geometric Stability Analysis for Differential Inclusions Governed by Maximally Monotone Operators
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917551497281536 |
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| author | Saoud, Hassan Théra, Michel Dao, Minh N. |
| author_facet | Saoud, Hassan Théra, Michel Dao, Minh N. |
| contents | This paper develops a geometric framework for the stability analysis of differential inclusions governed by maximally monotone operators. A key structural decomposition expresses the operator as the sum of a convexified limit mapping and a normal cone. However, the resulting dynamics are often difficult to analyze directly due to the absence of Lipschitz selections and boundedness. To overcome these challenges, we introduce a regularized system based on a fixed Lipschitz approximation of the convexified mapping. From this approximation, we extract a single-valued Lipschitz selection that preserves the essential geometric features of the original system. This framework enables the application of nonsmooth Lyapunov methods and Hamiltonian-based stability criteria. Instead of approximating trajectories, we focus on analyzing a simplified system that faithfully reflects the structure of the original dynamics. Several examples are provided to illustrate the method's practicality and scope. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_13000 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometric Stability Analysis for Differential Inclusions Governed by Maximally Monotone Operators Saoud, Hassan Théra, Michel Dao, Minh N. Optimization and Control This paper develops a geometric framework for the stability analysis of differential inclusions governed by maximally monotone operators. A key structural decomposition expresses the operator as the sum of a convexified limit mapping and a normal cone. However, the resulting dynamics are often difficult to analyze directly due to the absence of Lipschitz selections and boundedness. To overcome these challenges, we introduce a regularized system based on a fixed Lipschitz approximation of the convexified mapping. From this approximation, we extract a single-valued Lipschitz selection that preserves the essential geometric features of the original system. This framework enables the application of nonsmooth Lyapunov methods and Hamiltonian-based stability criteria. Instead of approximating trajectories, we focus on analyzing a simplified system that faithfully reflects the structure of the original dynamics. Several examples are provided to illustrate the method's practicality and scope. |
| title | Geometric Stability Analysis for Differential Inclusions Governed by Maximally Monotone Operators |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2507.13000 |