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Proper orthogonal decomposition based simultaneous approach for solving PDE-constrained optimal control problems
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DOI：10.1016/j.jprocont.2025.103392.png)
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Abstract
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Optimal control problems constrained by partial differential equations (PDEs) are widely encountered in engineering and scientific research and remain a challenging topic. Traditional methods for solving such problems typically discretize the temporal and spatial domains, often resulting in large-scale nonlinear programming problems that are computationally intensive to solve. This paper proposes a simultaneous approach based on proper orthogonal decomposition (POD). The method adopts a rolling strategy for simulating the PDEs and develops a reduced-order model using POD. To minimize the discrepancy between the reduced-order model and the full-order model, a heuristic strategy utilizing posterior error estimation is designed to enhance the model's accuracy. Numerical results indicate that the POD-based simultaneous approach not only substantially reduces the computational burden but also yields accurate solutions for PDE-constrained optimal control problems.
Keywords:
PDE-constrained optimization
Space-time orthogonal collocation on finite elements
Proper orthogonal decomposition
Model reduction
Simultaneous approach
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