Quadratic costs do not always work in MPC

We consider model predictive control (MPC) without terminal costs and constraints. Firstly, we rigorously show that MPC based on quadratic stage costs may fail, i.e., there does not exist a prediction horizon length such that a (controlled) equilibrium is asymptotically stable for the MPC closed loop although the system is, e.g., finite time controllable. Hence, stability properties of the infinite horizon optimal control problem are, in general, not preserved in MPC as long as purely quadratic costs are employed. This shows the necessity of using the stage cost as a design parameter to achieve asymptotic stability. Furthermore, we relax the standard controllability assumption employed in MPC without terminal costs and constraints to alleviate its verification.