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This paper continues the study (Yegorov, Dower, and Grüne et al.) and develops a curse‐of‐dimensionality‐free numerical approach to feedback stabilization, whose theoretical foundation was built in Yegorov et al. and involved the characterization of control Lyapunov functions (CLFs) via exit‐time optimal control. First, we describe an auxiliary linearization‐based technique for the construction of a local CLF and discuss how to determine its appropriate sublevel set that can serve as the terminal set in the exit‐time optimal control problem leading to a global or semi‐global CLF. Next, the curse of complexity is addressed with regard to the approximation of CLFs and associated feedback strategies in high‐dimensional regions. The goal is to enable for efficient model predictive control implementations with essentially faster (though less accurate) online policy updates than in case of solving direct or characteristics‐based nonlinear programming problems for each sample instant. We propose a computational approach that combines gradient enhanced modifications of the Kriging and inverse distance weighting frameworks for scattered grid interpolation. It in particular allows for convenient offline inclusion of new data to improve obtained approximations (machine learning can be used to select relevant new sparse grid nodes). Moreover, our method is designed so as to a priori return proper values of the CLF interpolant and its gradient on the entire terminal set of the considered exit‐time optimal control problem. Supporting numerical simulation results are also presented.

Given a continuous time nonlinear closed loop system, we give explicit constructions for sampled–data feedback laws for which the trajectories of the sampled– data closed loop system converge to the continuous time trajectories with a prescribed rate of convergence as the sampling interval length tends to zero. In particular, we investigate necessary and sufficient conditions under which such sampled–data feedbacks exist. We give analytic solutions to the problem for local orders of convergence _ 4 and present a MAPLE code for general orders.

Given a continuous-time controller and a Lyapunov function that shows global asymptotic stability for the closed-loop system, we provide several results for modification of the controller for sampled-data implementation. The main idea behind this approach is to use a particular structure for the redesigned controller and the main technical result is to show that the Fliess series expansions (in the sampling period T) of the Lyapunov difference for the sampled-data system with the redesigned controller have a very special form that is useful for controller redesign. We present results on controller redesign that achieve two different goals. The first goal is making the lower-order terms (in T) in the series expansion of the Lyapunov difference with the redesigned controller more negative. These control laws are very similar to those obtained from Lyapunov-based redesign of continuous-time systems for robustification of control laws and they often lead to corrections of the well-known “-LgV” form. The second goal is making the lower-order terms (in T) in the Fliess expansions of the Lyapunov difference for the sampled-data system with the redesigned controller behave as close as possible to the lower-order terms of the Lyapunov difference along solutions of the “ideal” sampled response of the continuous-time system with the original controller. In this case, the controller correction is very different from the first case and it contains appropriate “prediction” terms. The method is very flexible and one may try to achieve other objectives not addressed in this paper or derive similar results under different conditions. Simulation studies verify that redesigned controllers perform better (in an appropriate sense) than the unmodified ones when they are digitally implemented with sufficiently small sampling period T.