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RWTH Publication No: 841574 2021 
TITLE 
Reduced Order Model Predictive Control for Parametrized Parabolic Partial Differential Equations 
AUTHORS 
Saskia Dietze, Martin A. Grepl 
ABSTRACT 
Model Predictive Control (MPC) is a wellestablished approach to solve infinite horizon optimal control problems. Since optimization over an infinite time horizon is generally infeasible, MPC determines a suboptimal feedback control by repeatedly solving finite time optimal control problems. Although MPC has been successfully used in many applications, applying MPC to largescale systems  arising, e.g., through discretization of partial differential equations  requires the solution of highdimensional optimal control problems and thus poses immense computational effort.
We consider systems governed by parametrized parabolic partial differential equations and employ the reduced basis (RB) method as a lowdimensional surrogate model for the finite time optimal control problem. The reduced order optimal control serves as feedback control for the original largescale system. We analyze the proposed RBMPC approach by first developing a posteriori error bounds for the errors in the optimal control and associated cost functional. These bounds can be evaluated efficiently in an offlineonline computational procedure and allow us to guarantee asymptotic stability of the closedloop system using the RBMPC approach in several practical scenarios. We also propose an adaptive strategy to choose the prediction horizon of the finite time optimal control problem. Numerical results are presented to illustrate the theoretical properties of our approach. 
KEYWORDS 
model predictive control, suboptimality, asymptotic stability, reduced basis method, a posteriori error estimation, model
order reduction, partial differential equations, parameter dependent systems 
DOI 
10.1016/j.amc.2023.128044 
PUBLICATION 
Applied Mathematics and Computation, Volume 453, 15 September 2023, 128044 