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.

中文翻译：

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使用退出时间最优控制合成控制李雅普诺夫函数和稳定反馈策略第二部分：数值方法

本文继续研究（Yegorov、Dower 和 Grüne 等人）并开发了一种用于反馈稳定的无维数诅咒数值方法，其理论基础是由 Yegorov 等人建立的。并涉及通过退出时间最优控制来表征控制李雅普诺夫函数 (CLF)。首先，我们描述了一种用于构建局部 CLF 的基于辅助线性化的技术，并讨论如何确定其适当的子级集，该子级集可以作为导致全局或半全局 CLF 的退出时间最优控制问题中的终端集. 接下来，关于高维区域中 CLF 的近似和相关反馈策略的复杂性诅咒得到解决。目标是实现有效的模型预测控制实现，与解决每个样本时刻的直接或基于特征的非线性规划问题相比，在线策略更新速度明显更快（虽然不太准确）。我们提出了一种计算方法，该方法结合了克里金法的梯度增强修改和用于分散网格插值的逆距离加权框架。它特别允许方便地离线包含新数据以改进获得的近似值（机器学习可用于选择相关的新稀疏网格节点）。此外，我们的方法被设计为先验地返回 CLF 插值的正确值及其在所考虑的退出时间最优控制问题的整个终端集上的梯度。还提供了支持的数值模拟结果。