  Determining solving-time certificates of nonlinear model predictive control
(NMPC) implementations is a pressing requirement when deploying NMPC in
production environments. Such a certificate guarantees that the NMPC controller
returns a solution before the next sampling time. However, NMPC formulations
produce nonlinear programs (NLPs) for which it is very difficult to derive
their solving-time certificates. Our previous work, Wu and Braatz (2023),
challenged this limitation with a proposed input-constrained MPC algorithm
having exact iteration complexity but was restricted to linear MPC
formulations. This work extends the algorithm to solve input-constrained NMPC
problems, by using the Koopman operator and a condensing MPC technique. We
illustrate the algorithm performance on a high-dimensional, nonlinear partial
differential equation (PDE) control case study, in which we theoretically and
numerically certify the solving time to be less than the sampling time.
