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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/2412.08219 |
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| _version_ | 1866914065438212096 |
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| author | Qi, Jie Hu, Jiaqi Zhang, Jing Krstic, Miroslav |
| author_facet | Qi, Jie Hu, Jiaqi Zhang, Jing Krstic, Miroslav |
| contents | A transport PDE with a spatial integral and recirculation with constant delay has been a benchmark for neural operator approximations of PDE backstepping controllers. Introducing a spatially-varying delay into the model gives rise to a gain operator defined through integral equations which the operator's input -- the varying delay function -- enters in previously unencountered manners, including in the limits of integration and as the inverse of the `delayED time' function. This, in turn, introduces novel mathematical challenges in estimating the operator's Lipschitz constant. The backstepping kernel function having two branches endows the feedback law with a two-branch structure, where only one of the two feedback branches depends on both of the kernel branches. For this rich feedback structure, we propose a neural operator approximation of such a two-branch feedback law and prove the approximator to be semiglobally practically stabilizing. With numerical results we illustrate the training of the neural operator and its stabilizing capability. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_08219 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Neural Operator Feedback for a First-Order PIDE with Spatially-Varying State Delay Qi, Jie Hu, Jiaqi Zhang, Jing Krstic, Miroslav Systems and Control A transport PDE with a spatial integral and recirculation with constant delay has been a benchmark for neural operator approximations of PDE backstepping controllers. Introducing a spatially-varying delay into the model gives rise to a gain operator defined through integral equations which the operator's input -- the varying delay function -- enters in previously unencountered manners, including in the limits of integration and as the inverse of the `delayED time' function. This, in turn, introduces novel mathematical challenges in estimating the operator's Lipschitz constant. The backstepping kernel function having two branches endows the feedback law with a two-branch structure, where only one of the two feedback branches depends on both of the kernel branches. For this rich feedback structure, we propose a neural operator approximation of such a two-branch feedback law and prove the approximator to be semiglobally practically stabilizing. With numerical results we illustrate the training of the neural operator and its stabilizing capability. |
| title | Neural Operator Feedback for a First-Order PIDE with Spatially-Varying State Delay |
| topic | Systems and Control |
| url | https://arxiv.org/abs/2412.08219 |