Extremum-Seeking Boundary Control for Schrödinger-Type PDEs
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866918202487865344 |
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| author | Biazetto, Paulo Henrique Foganholo de Andrade, Gustavo Artur Oliveira, Tiago Roux Krstic, Miroslav |
| author_facet | Biazetto, Paulo Henrique Foganholo de Andrade, Gustavo Artur Oliveira, Tiago Roux Krstic, Miroslav |
| contents | This paper addresses the design and analysis of an extremum-seeking (ES) controller for scalar static maps in the context of infinite-dimensional dynamics governed by complex-valued partial differential equations (PDEs) of Schrodinger type. The system is actuated at one boundary, and the map input is defined as a real-valued quadratic functional corresponding to the squared norm of the complex state at the uncontrolled boundary. An isomorphism between the complex Hilbert space and its two-dimensional real-valued representation is established to enable the use of the standard multivariable Newton-based ES method. To compensate for the PDE actuation dynamics, a boundary control strategy based on a two-step backstepping procedure is employed. With a perturbation-based estimate of the Hessian inverse, the local exponential stability to a small neighborhood of the unknown extremum point is proved. A numerical example illustrates the effectiveness of the proposed extremum-seeking methodology. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_11994 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Extremum-Seeking Boundary Control for Schrödinger-Type PDEs Biazetto, Paulo Henrique Foganholo de Andrade, Gustavo Artur Oliveira, Tiago Roux Krstic, Miroslav Optimization and Control This paper addresses the design and analysis of an extremum-seeking (ES) controller for scalar static maps in the context of infinite-dimensional dynamics governed by complex-valued partial differential equations (PDEs) of Schrodinger type. The system is actuated at one boundary, and the map input is defined as a real-valued quadratic functional corresponding to the squared norm of the complex state at the uncontrolled boundary. An isomorphism between the complex Hilbert space and its two-dimensional real-valued representation is established to enable the use of the standard multivariable Newton-based ES method. To compensate for the PDE actuation dynamics, a boundary control strategy based on a two-step backstepping procedure is employed. With a perturbation-based estimate of the Hessian inverse, the local exponential stability to a small neighborhood of the unknown extremum point is proved. A numerical example illustrates the effectiveness of the proposed extremum-seeking methodology. |
| title | Extremum-Seeking Boundary Control for Schrödinger-Type PDEs |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2511.11994 |