Optimal Control of Parabolic Differential Equations Using Radau Collocation
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866910052133109760 |
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| author | Davies, Alexander M. Pollock, Sara Dennis, Miriam E. Rao, Anil V. |
| author_facet | Davies, Alexander M. Pollock, Sara Dennis, Miriam E. Rao, Anil V. |
| contents | A method is presented for the numerical solution of optimal boundary control problems governed by parabolic partial differential equations. The continuous space-time optimal control problem is transcribed into a sparse nonlinear programming problem through state and control parameterization. In particular, a multi-interval flipped Legendre-Gauss-Radau collocation method is implemented for temporal discretization alongside a Galerkin finite element spatial discretization. The finite element discretization allows for a reduction in problem size and avoids the redefinition of constraints required under a previous method. Further, a generalization of a Kirchoff transformation is performed to handle variational form nonlinearities in the context of numerical optimization. Due to the correspondence between the collocation points and the applied boundary conditions, the multi-interval flipped Legendre-Gauss-Radau collocation method is demonstrated to be preferable over the standard Legendre-Gauss-Radau collocation method for optimal control problems governed by parabolic partial differential equations. The details of the resulting transcription of the optimal control problem into a nonlinear programming problem are provided. Numerical examples demonstrate that the use of a multi-interval flipped Legendre-Gauss-Radau temporal discretization can lead to a reduction in the required number of collocation points to compute accurate values of the optimal objective in comparison to other methods. Lastly, a self-convergence analysis on each test problem illustrates that the error decays exponentially as a function of the mesh size in both the temporal and spatial dimensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_09815 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal Control of Parabolic Differential Equations Using Radau Collocation Davies, Alexander M. Pollock, Sara Dennis, Miriam E. Rao, Anil V. Optimization and Control 49M41, 49M37, 76D55, 80M50 A method is presented for the numerical solution of optimal boundary control problems governed by parabolic partial differential equations. The continuous space-time optimal control problem is transcribed into a sparse nonlinear programming problem through state and control parameterization. In particular, a multi-interval flipped Legendre-Gauss-Radau collocation method is implemented for temporal discretization alongside a Galerkin finite element spatial discretization. The finite element discretization allows for a reduction in problem size and avoids the redefinition of constraints required under a previous method. Further, a generalization of a Kirchoff transformation is performed to handle variational form nonlinearities in the context of numerical optimization. Due to the correspondence between the collocation points and the applied boundary conditions, the multi-interval flipped Legendre-Gauss-Radau collocation method is demonstrated to be preferable over the standard Legendre-Gauss-Radau collocation method for optimal control problems governed by parabolic partial differential equations. The details of the resulting transcription of the optimal control problem into a nonlinear programming problem are provided. Numerical examples demonstrate that the use of a multi-interval flipped Legendre-Gauss-Radau temporal discretization can lead to a reduction in the required number of collocation points to compute accurate values of the optimal objective in comparison to other methods. Lastly, a self-convergence analysis on each test problem illustrates that the error decays exponentially as a function of the mesh size in both the temporal and spatial dimensions. |
| title | Optimal Control of Parabolic Differential Equations Using Radau Collocation |
| topic | Optimization and Control 49M41, 49M37, 76D55, 80M50 |
| url | https://arxiv.org/abs/2505.09815 |