  The core of the Model Predictive Control (MPC) method in every step of the
algorithm consists in solving a time-dependent optimization problem on the
prediction horizon of the MPC algorithm, and then to apply a portion of the
optimal control over the application horizon to obtain the new state. To solve
this problem efficiently, we propose a time-adaptive residual a-posteriori
error control concept based on the optimality system of this optimal control
problem. This approach not only delivers a tailored time discretization of the
the prediction horizon, but also suggests a tailored length of the application
horizon for the current MPC step. We apply this concept for systems governed by
linear parabolic PDEs and present several numerical examples which demonstrate
the performance and the robustness of our adaptive MPC control concept.
