Dissipativity-based time domain decomposition for optimal control of hyperbolic PDEs
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866908444714336256 |
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| author | Farkas, Bálint Jacob, Birgit Schaller, Manuel Schmitz, Merlin |
| author_facet | Farkas, Bálint Jacob, Birgit Schaller, Manuel Schmitz, Merlin |
| contents | We propose a time domain decomposition approach to optimal control of partial differential equations (PDEs) based on semigroup theoretic methods. We formulate the optimality system consisting of two coupled forward-backward PDEs, the state and adjoint equation, as a sum of dissipative operators, which enables a Peaceman-Rachford-type fixed-point iteration. The iteration steps may be understood and implemented as solutions of many decoupled, and therefore highly parallelizable, time-distributed optimal control problems. We prove the convergence of the state, the control, and the corresponding adjoint state in function space. Due to the general framework of $C_0$-(semi)groups, the results are particularly well applicable, e.g., to hyperbolic equations, such as beam or wave equations. We illustrate the convergence and efficiency of the proposed method by means of two numerical examples subject to a 2D wave equation and a 3D heat equation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_07812 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dissipativity-based time domain decomposition for optimal control of hyperbolic PDEs Farkas, Bálint Jacob, Birgit Schaller, Manuel Schmitz, Merlin Optimization and Control Functional Analysis 46N10, 49M27, 49N10, 65M55, 65Y05 We propose a time domain decomposition approach to optimal control of partial differential equations (PDEs) based on semigroup theoretic methods. We formulate the optimality system consisting of two coupled forward-backward PDEs, the state and adjoint equation, as a sum of dissipative operators, which enables a Peaceman-Rachford-type fixed-point iteration. The iteration steps may be understood and implemented as solutions of many decoupled, and therefore highly parallelizable, time-distributed optimal control problems. We prove the convergence of the state, the control, and the corresponding adjoint state in function space. Due to the general framework of $C_0$-(semi)groups, the results are particularly well applicable, e.g., to hyperbolic equations, such as beam or wave equations. We illustrate the convergence and efficiency of the proposed method by means of two numerical examples subject to a 2D wave equation and a 3D heat equation. |
| title | Dissipativity-based time domain decomposition for optimal control of hyperbolic PDEs |
| topic | Optimization and Control Functional Analysis 46N10, 49M27, 49N10, 65M55, 65Y05 |
| url | https://arxiv.org/abs/2507.07812 |