# Optimal control for a class of linear transport-dominated systems via the shifted proper orthogonal decomposition

arXiv: https://arxiv.org/abs/2412.18950 · HTML: https://mlsanity.com/paper/2412.18950/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Dec 2024.

## Built on by (most cited first, 1 of 1)

- [Leveraging time and parameters for nonlinear model reduction methods](https://mlsanity.com/paper/2501.03853/) (2025)

## Abstract (arXiv)

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 high-fidelity, low-dimensional representations of transport-dominated phenomena. Furthermore, we propose two distinct frameworks for addressing these problems: one where the reduced-order model is constructed first, followed by optimization of the reduced system, and another where the original PDE system is optimized first, with the reduced-order model subsequently applied to the optimality system. We consider a 1D linear advection equation problem and compare the computational performance of the shifted POD method against the conventional methods like the standard POD when the reduced-order models are used as surrogates within a backtracking line search.