The Kalman rank condition and the optimal cost for the null-controllability of coupled Stokes systems
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866910816641482752 |
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| author | Balc'h, Kévin Le de Teresa, Luz |
| author_facet | Balc'h, Kévin Le de Teresa, Luz |
| contents | This paper considers the controllability of a class of coupled Stokes systems with distributed controls. The coupling terms are of a different nature. The first coupling is through the principal part of the Stokes operator with a constant real-valued positive-definite matrix. The second one acts through zero-order terms with a constant real-valued matrix. We assume the controls have their support in different measurable subsets of the spatial domain. Our main result states that such a system is small-time null-controllable if and only if a Kalman rank condition is satisfied. Moreover, when this condition holds, we prove the sharp upper bound for the cost of null-controllability for these systems. Our method is based on two ingredients. We start from the recent spectral estimate for the Stokes operator from Chaves-Silva, Souza, and Zhang. Then, we adapt Lissy and Zuazua's strategy concerning the internal observability for coupled systems of linear parabolic equations to coupled Stokes systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_03690 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Kalman rank condition and the optimal cost for the null-controllability of coupled Stokes systems Balc'h, Kévin Le de Teresa, Luz Optimization and Control 76D05, 35Q30, 93B05, 93B07, 93C10 This paper considers the controllability of a class of coupled Stokes systems with distributed controls. The coupling terms are of a different nature. The first coupling is through the principal part of the Stokes operator with a constant real-valued positive-definite matrix. The second one acts through zero-order terms with a constant real-valued matrix. We assume the controls have their support in different measurable subsets of the spatial domain. Our main result states that such a system is small-time null-controllable if and only if a Kalman rank condition is satisfied. Moreover, when this condition holds, we prove the sharp upper bound for the cost of null-controllability for these systems. Our method is based on two ingredients. We start from the recent spectral estimate for the Stokes operator from Chaves-Silva, Souza, and Zhang. Then, we adapt Lissy and Zuazua's strategy concerning the internal observability for coupled systems of linear parabolic equations to coupled Stokes systems. |
| title | The Kalman rank condition and the optimal cost for the null-controllability of coupled Stokes systems |
| topic | Optimization and Control 76D05, 35Q30, 93B05, 93B07, 93C10 |
| url | https://arxiv.org/abs/2502.03690 |