Two-stage model reduction approaches for the efficient and certified solution of parametrized optimal control problems

TL;DR

This paper introduces a two-stage reduced order modeling approach for parametrized linear-quadratic optimal control problems, significantly improving computational efficiency while maintaining accuracy, and providing reliable error bounds.

Contribution

It develops a novel combined reduced basis approach for both system dynamics and adjoint states, with proven error estimation, tailored for efficient solutions of parametrized optimal control problems.

Findings

Achieves computational complexity independent of original state space size

Provides reliable a posteriori error bounds for the reduced models

Demonstrates significant speedup over exact solutions in numerical experiments

Abstract

In this contribution we develop an efficient reduced order model for solving parametrized linear-quadratic optimal control problems with linear time-varying state system. The fully reduced model combines reduced basis approximations of the system dynamics and of the manifold of optimal final time adjoint states to achieve a computational complexity independent of the original state space. Such a combination is particularly beneficial in the case where a deviation in a low-dimensional output is penalized. In addition, an offline-online decomposed a posteriori error estimator bounding the error between the approximate final time adjoint with respect to the optimal one is derived and its reliability proven. We propose different strategies for building the involved reduced order models, for instance by separate reduction of the dynamical systems and the final time adjoint states or via greedy procedures yielding a combined and fully reduced model. These algorithms are evaluated and compared for a two-dimensional heat equation problem. The numerical results show the desired accuracy of the reduced models and highlight the speedup obtained by the newly combined reduced order model in comparison to an exact computation of the optimal control or other reduction approaches.