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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Online-Zugang: | https://arxiv.org/abs/2407.07446 |
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| _version_ | 1866915322721730560 |
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| author | Gherdaoui, Théo |
| author_facet | Gherdaoui, Théo |
| contents | The goal of this article is to contribute to a better understanding of the relations between the exact controllability of nonlinear PDEs and the control theory for ODEs based on Lie brackets, through a study of the Schrödinger PDE with bilinear control. We focus on the small-time local controllability (STLC) around an equilibrium, when the linearized system is not controllable. We study the second-order term in the Taylor expansion of the state, with respect to the control. For scalar-input ODEs, quadratic terms never recover controllability: they induce signed drifts in the dynamics. Thus proving STLC requires to go at least to the third order. Similar results were proved for the bilinear Schrödinger PDE with scalar-input controls. In this article, we study the case of multi-input systems. We clarify among the quadratic Lie brackets, those that allow to recover STLC: they are bilinear with respect to two different controls. For ODEs, our result is a consequence of Sussman's sufficient condition $S(θ)$ (when focused on quadratic terms), but we propose a new proof, designed to prepare an easier transfer to PDEs. This proof relies on a representation formula of the state inspired by the Magnus formula. By adapting it, we prove a new STLC result for the bilinear Schrödinger PDE. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_07446 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Small-Time Local Controllability of the multi-input bilinear Schrödinger equation thanks to a quadratic term Gherdaoui, Théo Optimization and Control The goal of this article is to contribute to a better understanding of the relations between the exact controllability of nonlinear PDEs and the control theory for ODEs based on Lie brackets, through a study of the Schrödinger PDE with bilinear control. We focus on the small-time local controllability (STLC) around an equilibrium, when the linearized system is not controllable. We study the second-order term in the Taylor expansion of the state, with respect to the control. For scalar-input ODEs, quadratic terms never recover controllability: they induce signed drifts in the dynamics. Thus proving STLC requires to go at least to the third order. Similar results were proved for the bilinear Schrödinger PDE with scalar-input controls. In this article, we study the case of multi-input systems. We clarify among the quadratic Lie brackets, those that allow to recover STLC: they are bilinear with respect to two different controls. For ODEs, our result is a consequence of Sussman's sufficient condition $S(θ)$ (when focused on quadratic terms), but we propose a new proof, designed to prepare an easier transfer to PDEs. This proof relies on a representation formula of the state inspired by the Magnus formula. By adapting it, we prove a new STLC result for the bilinear Schrödinger PDE. |
| title | Small-Time Local Controllability of the multi-input bilinear Schrödinger equation thanks to a quadratic term |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2407.07446 |