Estimates of the Kolmogorov n-width for nonlinear transformations with application to distributed-parameter control systems
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910507269619712 |
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| author | Zuyev, Alexander Feng, Lihong Benner, Peter |
| author_facet | Zuyev, Alexander Feng, Lihong Benner, Peter |
| contents | This paper aims at characterizing the approximability of bounded sets in the range of nonlinear operators in Banach spaces by finite-dimensional linear varieties. In particular, the class of operators we consider includes the endpoint maps of nonlinear distributed-parameter control systems. We describe the relationship between the Kolmogorov n-width of a bounded subset and the width of its image under an essentially nonlinear transformation. We propose explicit estimates of the n-width in the space of images in terms of the affine part of the corresponding operator and the width of its nonlinear perturbation. These $n$-width estimates enable us to describe the reachable sets for infinite-dimensional bilinear control systems, with applications to controlling the Euler-Bernoulli beam using a contraction force and to a single-input Schrödinger equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_06029 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Estimates of the Kolmogorov n-width for nonlinear transformations with application to distributed-parameter control systems Zuyev, Alexander Feng, Lihong Benner, Peter Optimization and Control 93C20, 93C25, 41A25, 41A46, 81Q93, 74K10 This paper aims at characterizing the approximability of bounded sets in the range of nonlinear operators in Banach spaces by finite-dimensional linear varieties. In particular, the class of operators we consider includes the endpoint maps of nonlinear distributed-parameter control systems. We describe the relationship between the Kolmogorov n-width of a bounded subset and the width of its image under an essentially nonlinear transformation. We propose explicit estimates of the n-width in the space of images in terms of the affine part of the corresponding operator and the width of its nonlinear perturbation. These $n$-width estimates enable us to describe the reachable sets for infinite-dimensional bilinear control systems, with applications to controlling the Euler-Bernoulli beam using a contraction force and to a single-input Schrödinger equation. |
| title | Estimates of the Kolmogorov n-width for nonlinear transformations with application to distributed-parameter control systems |
| topic | Optimization and Control 93C20, 93C25, 41A25, 41A46, 81Q93, 74K10 |
| url | https://arxiv.org/abs/2403.06029 |