Nominal stability of real-time iteration scheme for nonlinear model predictive control

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The advanced-step NMPC controller: Optimality, stability and robustness (2009, background): “Under mild conditions, if the sensitivity approximation is used to warm-start the next problem, the background problem can be converged in a few iterations (Diehl et al., 2005).”
From linear to nonlinear MPC: bridging the gap via the real-time iteration (2020, methodology): “It has been formally studied in Diehl, Findeisen, Allgöwer, Bock, and Schlöder (2005), and has been veri fi ed in many deployments of the RTI approach.”
Fast Direct Multiple Shooting Algorithms for Optimal Robot Control (2005, background): “The scheme’s theoretical contraction properties have been investigated in [21] for the variant described in this paper, and in [22, 19] for other variants.”
ACADO toolkit—An open‐source framework for automatic control and dynamic optimization (2011, background): “Error bounds and closed-loop stability of the scheme have been established for nonlinear MPC with shifted and non-shifted initializations in [29, 30].”
Efficient Numerical Methods for Nonlinear MPC and Moving Horizon Estimation (2009, background): “As a sequential scheme without tangential predictor, it seems to have had sometimes problems with nonlinear convergence, though closed-loop stability was proven for open-loop stable processes [41], and in principle, the theoretical NMPC stability analysis from [18] is applicable.”