  In this technical note we analyse the performance improvement and optimality
properties of the Learning Model Predictive Control (LMPC) strategy for linear
deterministic systems. The LMPC framework is a policy iteration scheme where
closed-loop trajectories are used to update the control policy for the next
execution of the control task. We show that, when a Linear Independence
Constraint Qualification (LICQ) condition holds, the LMPC scheme guarantees
strict iterative performance improvement and optimality, meaning that the
closed-loop cost evaluated over the entire task converges asymptotically to the
optimal cost of the infinite-horizon control problem. Compared to previous
works this sufficient LICQ condition can be easily checked, it holds for a
larger class of systems and it can be used to adaptively select the prediction
horizon of the controller, as demonstrated by a numerical example.
