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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.23708 |
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| _version_ | 1866918407511736320 |
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| author | Freund, Anton Pischke, Nicholas |
| author_facet | Freund, Anton Pischke, Nicholas |
| contents | We provide quantitative convergence results for continuous-time dynamical systems in metric spaces that satisfy a continuous-time analog of quasi-Fejér monotonicity. More precisely, we provide a (strong) convergence result for such dynamical systems over compact metric spaces which is quantitatively outfitted with a continuous-time rate of metastability, which moreover can be explicitly and effectively constructed in a very uniform way, only depending on a few moduli representing quantitative witnesses to key properties of the dynamical system and a measure for the compactness of the space. We further show how this convergence result can be extended to non-compact spaces under a regularity assumption of the associated problem, where moreover rates of convergence can then be explicitly constructed which are similarly uniform. In both cases, already the associated ``infinitary'' convergence result is qualitatively novel in its present generality. Beyond this abstract quantitative theory for such dynamical systems, we motivate how the presently studied continuous-time variant of quasi-Fejér monotonicity naturally occurs as a unifying property of many dynamical systems and differential equations and inclusions, and in that way can be used to provide a comprehensive quantitative theory for many such dynamical systems. We illustrate this with three case studies for both classical first- and second-order dynamical systems in Hilbert spaces as well as (generalized) gradient flows and associated semigroups in nonlinear Hadamard spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_23708 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Effective rates for continuous-time quasi-Fejér monotone dynamical systems Freund, Anton Pischke, Nicholas Optimization and Control We provide quantitative convergence results for continuous-time dynamical systems in metric spaces that satisfy a continuous-time analog of quasi-Fejér monotonicity. More precisely, we provide a (strong) convergence result for such dynamical systems over compact metric spaces which is quantitatively outfitted with a continuous-time rate of metastability, which moreover can be explicitly and effectively constructed in a very uniform way, only depending on a few moduli representing quantitative witnesses to key properties of the dynamical system and a measure for the compactness of the space. We further show how this convergence result can be extended to non-compact spaces under a regularity assumption of the associated problem, where moreover rates of convergence can then be explicitly constructed which are similarly uniform. In both cases, already the associated ``infinitary'' convergence result is qualitatively novel in its present generality. Beyond this abstract quantitative theory for such dynamical systems, we motivate how the presently studied continuous-time variant of quasi-Fejér monotonicity naturally occurs as a unifying property of many dynamical systems and differential equations and inclusions, and in that way can be used to provide a comprehensive quantitative theory for many such dynamical systems. We illustrate this with three case studies for both classical first- and second-order dynamical systems in Hilbert spaces as well as (generalized) gradient flows and associated semigroups in nonlinear Hadamard spaces. |
| title | Effective rates for continuous-time quasi-Fejér monotone dynamical systems |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2603.23708 |