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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.18641 |
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| _version_ | 1866910016957579264 |
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| author | Zamorano, Sebastián Zuazua, Enrique |
| author_facet | Zamorano, Sebastián Zuazua, Enrique |
| contents | We develop a functional-analytic characterization of output tracking controllability for finite-dimensional linear systems. By formulating tracking as the surjectivity of the control-to-output map on suitable trajectory spaces, we show that exact tracking is equivalent to a nonstandard observability inequality for the adjoint dynamics. This characterization enables a Hilbert Uniqueness Method (HUM) type variational construction of minimum-norm tracking controls and makes explicit the intrinsic regularity requirements on reference trajectories induced by the system dynamics and the output operator. The same framework also yields a natural notion of approximate tracking when exact tracking fails. We provide explicit formulas in the scalar case and report numerical experiments for ODEs and semi-discretized PDEs, demonstrating the method for both smooth and non-smooth targets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_18641 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tracking controllability for finite-dimensional linear systems Zamorano, Sebastián Zuazua, Enrique Optimization and Control Classical Analysis and ODEs We develop a functional-analytic characterization of output tracking controllability for finite-dimensional linear systems. By formulating tracking as the surjectivity of the control-to-output map on suitable trajectory spaces, we show that exact tracking is equivalent to a nonstandard observability inequality for the adjoint dynamics. This characterization enables a Hilbert Uniqueness Method (HUM) type variational construction of minimum-norm tracking controls and makes explicit the intrinsic regularity requirements on reference trajectories induced by the system dynamics and the output operator. The same framework also yields a natural notion of approximate tracking when exact tracking fails. We provide explicit formulas in the scalar case and report numerical experiments for ODEs and semi-discretized PDEs, demonstrating the method for both smooth and non-smooth targets. |
| title | Tracking controllability for finite-dimensional linear systems |
| topic | Optimization and Control Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2407.18641 |