  In this paper, an asymptotic stability proof for a class of methods for
inexact nonlinear model predictive control is presented. General Q-linearly
convergent online optimization methods are considered and an asymptotic
stability result is derived for the setting where a limited number of
iterations of the optimizer are carried out per sampling time. Under the
assumption of Lipschitz continuity of the solution, we explicitly construct a
Lyapunov function for the combined system-optimizer dynamics, which shows that
asymptotic stability can be obtained if the sampling time is sufficiently
short. The results constitute an extension to existing attractivity results
which hold in the simplified setting where inequality constraints are either
not present or inactive in the region of attraction considered. Moreover, with
respect to the established results on robust asymptotic stability of suboptimal
model predictive control, we develop a framework that takes into account the
optimizer's dynamics and does not require decrease of the objective function
across iterates. By extending these results, the gap between theory and
practice of the standard real-time iteration strategy is bridged and asymptotic
stability for a broader class of methods is guaranteed.
