Stabilizing stochastic MPC without terminal constraints

The stability proofs of Model Predictive Control without terminal constraints and/or cost are tightly based upon the principle of optimality, which does not hold in most currently employed approaches to Stochastic MPC. In this paper, we first provide a stability proof for Stochastic Model Predictive Control without terminal cost or constraints under the assumption of optimization over feedback laws and propagation of the probability density functions of predicted states. Based thereon, we highlight why the proof does not remain valid if approximations such as parametrized feedback laws or relaxations on the chance constraints are employed and provide tightened assumptions that are sufficient to establish closed-loop stability. General statements valid for nonlinear systems are provided along with examples and computational simplifications in the case of linear systems.