Boundary Effects on the Controllability of Coupled KdV Systems
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913725856874496 |
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| author | Gallego, F. A. Pazoto, A. F. Rivas, I. |
| author_facet | Gallego, F. A. Pazoto, A. F. Rivas, I. |
| contents | We study the exact boundary controllability of a nonlinear coupled system of two Korteweg-de Vries equations on a bounded interval. The model describes the interactions of two weakly nonlinear gravity waves in a stratified fluid. Due to the nature of the system, six boundary conditions are required. However, to study the controllability property, we consider a different combination of the control inputs, with a maximum of four. Firstly, the results are obtained for the linearized system through a classical duality approach and some hidden regularity properties of the boundary terms. This approach reduces the controllability problem to the study of a spectral problem, which is solved by using the Paley-Wiener method introduced by Rosier. Then, the issue is to establish when a certain quotient of entire functions still turns out to be an entire function. It can be viewed as a problem of factoring an entire function that, depending on the control configuration, leads to the study of a transcendental equation. Finally, by using the contraction mapping theorem, we derive the local controllability for the full system. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2302_13443 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Boundary Effects on the Controllability of Coupled KdV Systems Gallego, F. A. Pazoto, A. F. Rivas, I. Analysis of PDEs Optimization and Control 35Q53, 93B05, 37K10, 30D20 We study the exact boundary controllability of a nonlinear coupled system of two Korteweg-de Vries equations on a bounded interval. The model describes the interactions of two weakly nonlinear gravity waves in a stratified fluid. Due to the nature of the system, six boundary conditions are required. However, to study the controllability property, we consider a different combination of the control inputs, with a maximum of four. Firstly, the results are obtained for the linearized system through a classical duality approach and some hidden regularity properties of the boundary terms. This approach reduces the controllability problem to the study of a spectral problem, which is solved by using the Paley-Wiener method introduced by Rosier. Then, the issue is to establish when a certain quotient of entire functions still turns out to be an entire function. It can be viewed as a problem of factoring an entire function that, depending on the control configuration, leads to the study of a transcendental equation. Finally, by using the contraction mapping theorem, we derive the local controllability for the full system. |
| title | Boundary Effects on the Controllability of Coupled KdV Systems |
| topic | Analysis of PDEs Optimization and Control 35Q53, 93B05, 37K10, 30D20 |
| url | https://arxiv.org/abs/2302.13443 |