Exact output tracking for the one-dimensional heat equation and applications to the interpolation problem in Gevrey classes of order 2
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| Format: | Preprint |
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2025
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| author | Davron, Lucas Lissy, Pierre |
| author_facet | Davron, Lucas Lissy, Pierre |
| contents | This paper provides a complete characterization of the Dirichlet boundary outputs that can be exactly tracked in the one-dimensional heat equation with Neumann boundary control. The problem consists in describing the set of boundary traces generated by square-integrable controls over a finite or infinite time horizon. We show that these outputs form a precise functional space related to Gevrey regularity of order 2. In the infinite-time case, the trackable outputs are precisely those functions whose successive derivatives satisfy a weighted summability condition, which corresponds to specific Gevrey classes. For finite-time horizons, an additional compatibility condition involving the reachable space of the system provides a full characterization. The analysis relies on Fourier-Laplace transform, properties of Hardy spaces, the flatness method, and a new Plancherel-type theorem for Hilbert spaces of Gevrey functions. Beyond control theory, our results yield an optimal solution to the classical interpolation problem in Gevrey-$2$ classes, which improves results of Mitjagin on the optimal loss factor. The techniques developed here also extend to variants of the heat system with different boundary conditions or observation points. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_07326 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exact output tracking for the one-dimensional heat equation and applications to the interpolation problem in Gevrey classes of order 2 Davron, Lucas Lissy, Pierre Optimization and Control 26E10, 93C20, 35K05, 46EE10 This paper provides a complete characterization of the Dirichlet boundary outputs that can be exactly tracked in the one-dimensional heat equation with Neumann boundary control. The problem consists in describing the set of boundary traces generated by square-integrable controls over a finite or infinite time horizon. We show that these outputs form a precise functional space related to Gevrey regularity of order 2. In the infinite-time case, the trackable outputs are precisely those functions whose successive derivatives satisfy a weighted summability condition, which corresponds to specific Gevrey classes. For finite-time horizons, an additional compatibility condition involving the reachable space of the system provides a full characterization. The analysis relies on Fourier-Laplace transform, properties of Hardy spaces, the flatness method, and a new Plancherel-type theorem for Hilbert spaces of Gevrey functions. Beyond control theory, our results yield an optimal solution to the classical interpolation problem in Gevrey-$2$ classes, which improves results of Mitjagin on the optimal loss factor. The techniques developed here also extend to variants of the heat system with different boundary conditions or observation points. |
| title | Exact output tracking for the one-dimensional heat equation and applications to the interpolation problem in Gevrey classes of order 2 |
| topic | Optimization and Control 26E10, 93C20, 35K05, 46EE10 |
| url | https://arxiv.org/abs/2511.07326 |