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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.28442 |
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| _version_ | 1866917369396330496 |
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| author | Breiten, Tobias Burela, Shubhaditya Schulze, Philipp |
| author_facet | Breiten, Tobias Burela, Shubhaditya Schulze, Philipp |
| contents | Solving optimal control problems for transport-dominated partial differential equations (PDEs) can become computationally expensive, especially when dealing with high-dimensional systems. To overcome this challenge, we focus on developing and deriving reduced-order models that can replace the full PDE system in solving the optimal control problem. Specifically, we explore the use of the shifted proper orthogonal decomposition (POD) as a reduced-order model, which is particularly effective for capturing low-dimensional representations of high-fidelity transport-dominated phenomena. In this work, a reduced-order model is constructed first, followed by the optimization of the reduced system. We consider a 1D linear advection equation problem and prove existence and uniqueness of solutions for the reduced-order model as well as the existence of an optimal control. Moreover, we compare the computational performance of the shifted POD method against the standard POD. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_28442 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Optimal control with the shifted proper orthogonal decomposition via a first-reduce-then-optimize framework Breiten, Tobias Burela, Shubhaditya Schulze, Philipp Optimization and Control Solving optimal control problems for transport-dominated partial differential equations (PDEs) can become computationally expensive, especially when dealing with high-dimensional systems. To overcome this challenge, we focus on developing and deriving reduced-order models that can replace the full PDE system in solving the optimal control problem. Specifically, we explore the use of the shifted proper orthogonal decomposition (POD) as a reduced-order model, which is particularly effective for capturing low-dimensional representations of high-fidelity transport-dominated phenomena. In this work, a reduced-order model is constructed first, followed by the optimization of the reduced system. We consider a 1D linear advection equation problem and prove existence and uniqueness of solutions for the reduced-order model as well as the existence of an optimal control. Moreover, we compare the computational performance of the shifted POD method against the standard POD. |
| title | Optimal control with the shifted proper orthogonal decomposition via a first-reduce-then-optimize framework |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2603.28442 |