  Model Predictive Control (MPC) is a well-established 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
large-scale systems -- arising, e.g., through discretization of partial
differential equations -- requires the solution of high-dimensional 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 low-dimensional
surrogate model for the finite time optimal control problem. The reduced order
optimal control serves as feedback control for the original large-scale system.
We analyze the proposed RB-MPC 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 offline-online computational
procedure and allow us to guarantee asymptotic stability of the closed-loop
system using the RB-MPC 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.
