Optimal control with the shifted proper orthogonal decomposition via a first-reduce-then-optimize framework
Abstract
Solving optimal control problems for transport-dominated partial differential equations (PDEs) can become computationally expensive, especially when dealing with high-dimensional systems. To overcome this challenge, we focus on developing and deriving reduced-order models that can replace the full PDE system in solving the optimal control problem. Specifically, we explore the use of the shifted proper orthogonal decomposition (POD) as a reduced-order model, which is particularly effective for capturing low-dimensional representations of high-fidelity transport-dominated phenomena. In this work, a reduced-order model is constructed first, followed by the optimization of the reduced system. We consider a 1D linear advection equation problem and prove existence and uniqueness of solutions for the reduced-order model as well as the existence of an optimal control. Moreover, we compare the computational performance of the shifted POD method against the standard POD.
Cite
@article{arxiv.2603.28442,
title = {Optimal control with the shifted proper orthogonal decomposition via a first-reduce-then-optimize framework},
author = {Tobias Breiten and Shubhaditya Burela and Philipp Schulze},
journal= {arXiv preprint arXiv:2603.28442},
year = {2026}
}