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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.22468 |
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| _version_ | 1866913864170340352 |
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| author | Akian, Marianne Gaubert, Stéphane Marchesini, Loïc Morris, Ian |
| author_facet | Akian, Marianne Gaubert, Stéphane Marchesini, Loïc Morris, Ian |
| contents | The competitive spectral radius extends the notion of joint spectral radius to the two-player case: two players alternatively select matrices in prescribed compact sets, resulting in an infinite matrix product; one player wishes to maximize the growth rate of this product, whereas the other player wishes to minimize it. We show that when the matrices represent linear operators preserving a cone and satisfying a "strict positivity" assumption, the competitive spectral radius depends continuously - and even in a Lipschitz-continuous way - on the matrix sets. Moreover, we show that the competive spectral radius can be approximated up to any accuracy. This relies on the solution of a discretized infinite dimensional non-linear eigenproblem. We illustrate the approach with an example of age-structured population dynamics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_22468 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Continuity and approximability of competitive spectral radii Akian, Marianne Gaubert, Stéphane Marchesini, Loïc Morris, Ian Optimization and Control Numerical Analysis Dynamical Systems The competitive spectral radius extends the notion of joint spectral radius to the two-player case: two players alternatively select matrices in prescribed compact sets, resulting in an infinite matrix product; one player wishes to maximize the growth rate of this product, whereas the other player wishes to minimize it. We show that when the matrices represent linear operators preserving a cone and satisfying a "strict positivity" assumption, the competitive spectral radius depends continuously - and even in a Lipschitz-continuous way - on the matrix sets. Moreover, we show that the competive spectral radius can be approximated up to any accuracy. This relies on the solution of a discretized infinite dimensional non-linear eigenproblem. We illustrate the approach with an example of age-structured population dynamics. |
| title | Continuity and approximability of competitive spectral radii |
| topic | Optimization and Control Numerical Analysis Dynamical Systems |
| url | https://arxiv.org/abs/2505.22468 |