Proper orthogonal decomposition based simultaneous approach for solving PDE-constrained optimal control problems

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.