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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/2502.15532 |
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| _version_ | 1866912240270049280 |
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| author | Hartman, Andreas Koenig, Armand |
| author_facet | Hartman, Andreas Koenig, Armand |
| contents | In this paper we investigate null-controllable initial states of the half heat equation controlled from a sub-arc $ω$ of the unit circle. We also study the projection on positive frequencies of the half-heat equation. For this projected half-heat equation, we obtain necessary as well as sufficient conditions for an initial condition to be null-controllable. These conditions, which are almost sharp, are expressed in term of projections on positive frequencies of functions supported on $ω$. From these results, and with the help of classical results on sum of holomorphic and anti-holomorphic functions, we also treat the (unprojected) half-heat equation. Surprisingly, without using our conditions on null-controllable states, we are able to show that the space of null-controllable functions does not depend on time by using a result of separation of singularities for holomorphic functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_15532 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Control of the half-heat equation Hartman, Andreas Koenig, Armand Analysis of PDEs Complex Variables Optimization and Control 93C05, 93C20, 35R11, 30H10, 30H20, 30J99 In this paper we investigate null-controllable initial states of the half heat equation controlled from a sub-arc $ω$ of the unit circle. We also study the projection on positive frequencies of the half-heat equation. For this projected half-heat equation, we obtain necessary as well as sufficient conditions for an initial condition to be null-controllable. These conditions, which are almost sharp, are expressed in term of projections on positive frequencies of functions supported on $ω$. From these results, and with the help of classical results on sum of holomorphic and anti-holomorphic functions, we also treat the (unprojected) half-heat equation. Surprisingly, without using our conditions on null-controllable states, we are able to show that the space of null-controllable functions does not depend on time by using a result of separation of singularities for holomorphic functions. |
| title | Control of the half-heat equation |
| topic | Analysis of PDEs Complex Variables Optimization and Control 93C05, 93C20, 35R11, 30H10, 30H20, 30J99 |
| url | https://arxiv.org/abs/2502.15532 |