POD-Galerkin Model Order Reduction for Parametrized Time Dependent Linear Quadratic Optimal Control Problems in Saddle Point Formulation

TL;DR

This paper develops a POD-Galerkin reduced order modeling approach for parametrized time-dependent linear quadratic optimal control problems in saddle point form, enabling faster and accurate simulations.

Contribution

It introduces a saddle point formulation for parametrized optimal control problems and applies POD-Galerkin reduction to improve computational efficiency.

Findings

Effective reduction in computational time demonstrated

Applicable to boundary and distributed control problems

Validated on Graetz flow and Stokes equations

Abstract

In this work we recast parametrized time dependent optimal control problems governed by partial differential equations in a saddle point formulation and we propose reduced order methods as an effective strategy to solve them. Indeed, on one hand parametrized time dependent optimal control is a very powerful mathematical model which is able to describe several physical phenomena; on the other hand, it requires a huge computational effort. Reduced order methods are a suitable approach to have rapid and accurate simulations. We rely on POD-Galerkin reduction over the physical and geometrical parameters of the optimality system in a space-time formulation. Our theoretical results and our methodology are tested on two examples: a boundary time dependent optimal control for a Graetz flow and a distributed optimal control governed by time dependent Stokes equations. With these two experiments the convenience of the reduced order modelling is further extended to the field of time dependent optimal control.