This paper investigates the numerical approximation of infinite horizon optimal control problems, aiming to mitigate the curse of dimensionality by employing a novel Proper Orthogonal Decomposition (POD) method based on time derivatives.
The authors utilize a dynamic programming approach to approximate the value function of the optimal control problem, which solves a Hamilton-Jacobi-Bellman (HJB) equation. They propose a POD method that incorporates time derivatives into the snapshot selection process, leading to a reduced-order model of the original problem. The error analysis of the method leverages recently established optimal bounds for fully discrete approximations of HJB equations.
The paper concludes that incorporating time derivatives into the POD framework significantly enhances the accuracy and efficiency of solving infinite horizon optimal control problems. The proposed method effectively reduces the computational burden associated with high-dimensional problems while maintaining optimal convergence properties.
This research contributes to the field of numerical optimal control by introducing a novel and efficient POD-based reduced-order modeling technique. The use of time derivatives in snapshot selection and the rigorous error analysis provide a solid foundation for applying this method to complex, high-dimensional control problems arising in various engineering and scientific disciplines.
While the paper focuses on infinite horizon problems, extending the proposed method to finite horizon scenarios and exploring its applicability to stochastic optimal control problems represent promising avenues for future research.
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by Javier de Fr... at arxiv.org 11-06-2024
https://arxiv.org/pdf/2310.10552.pdfDeeper Inquiries