On Relaxing the <i>N</i>-Reachability Implicit Requirement in NMPC Design

This paper proposes a proof of stability for Model Predictive Control formulations involving a prediction horizon that might be too short to meet the reachability condition generally invoked as a sufficient condition for closed-loop stability. This condition is replaced by a contraction condition on the stage cost. But unlike the contraction based existing formulations where the prediction horizon becomes a decision variable, the formulation proposed in this paper remains standard in that it uses constant and short prediction horizon. An illustrative example is provided to assess the relevance of the proposed formulation.