A data-driven partitioned approach for the resolution of time-dependent optimal control problems with dynamic mode decomposition

TL;DR

This paper introduces a data-driven approach using Dynamic Mode Decomposition to efficiently solve time-dependent optimal control problems governed by PDEs, reducing computational costs while maintaining accuracy.

Contribution

It develops a novel DMD-based framework for recasting and solving PDE-governed optimal control problems with improved efficiency and predictive capabilities.

Findings

Effective reconstruction and prediction of system variables

Reduced computational cost compared to standard methods

Validated on flow and Stokes control problems

Abstract

This work recasts time-dependent optimal control problems governed by partial differential equations in a Dynamic Mode Decomposition with control framework. Indeed, since the numerical solution of such problems requires a lot of computational effort, we rely on this specific data-driven technique, using both solution and desired state measurements to extract the underlying system dynamics. Thus, after the Dynamic Mode Decomposition operators construction, we reconstruct and perform future predictions for all the variables of interest at a lower computational cost with respect to the standard space-time discretized models. We test the methodology in terms of relative reconstruction and prediction errors on a boundary control for a Graetz flow and on a distributed control with Stokes constraints.