Efficient Model Predictive Control for Parabolic PDEs with Goal Oriented Error Estimation

TL;DR

This paper introduces a goal-oriented error estimation approach to improve the efficiency of Model Predictive Control for parabolic PDEs by enabling adaptive discretization tailored to the control horizon.

Contribution

It develops a novel a posteriori error estimation method specifically designed for MPC, with proven exponential decay of error indicators, enhancing computational efficiency.

Findings

Error indicators decay exponentially outside the control support

Adaptive discretization improves computational efficiency in MPC

Numerical examples demonstrate the effectiveness of the approach

Abstract

We show how a posteriori goal oriented error estimation can be used to efficiently solve the subproblems occurring in a Model Predictive Control (MPC) algorithm. In MPC, only an initial part of a computed solution is implemented as a feedback, which motivates grid refinement particularly tailored to this context. To this end, we present a truncated cost functional as objective for goal oriented adaptivity and prove under stabilizability assumptions that error indicators decay exponentially outside the support of this quantity. This leads to very efficient time and space discretizations for MPC, which we will illustrate by means of various numerical examples.