Analyzing nonlinear systems with stabilizable controlled invariant sets (CISs) requires accurate estimation of their domains of stabilization (DOS) together with associated stabilizing controllers. Despite extensive research, estimating DOSs for general nonlinear systems remains challenging due to fundamental theoretical and computational limitations. In this paper, we propose a novel framework for estimating DOSs for controlled input-constrained discrete-time systems. The DOS is characterized via newly introduced value functions defined on metric spaces of compact sets. We establish the fundamental properties of these value functions and derive the associated Bellman-type (Zubov-type) functional equations. Building on this characterization, we develop a physics-informed neural network (NN) framework that learns the value functions by embedding the derived functional equations directly into the training process. The proposed methodology is demonstrated through two numerical examples, illustrating its ability to accurately estimate DOSs and synthesize stabilizing controllers from the learned value functions.

翻译：分析具有可镇定受控不变集（CIS）的非线性系统需要精确估计其稳定域（DOS）及相关镇定控制器。尽管已有大量研究，但由于基础理论与计算限制，估计一般非线性系统的DOS仍具挑战性。本文提出一种新型框架，用于估计受控输入约束离散时间系统的DOS。该DOS通过新引入的、定义于紧集度量空间上的值函数进行表征。我们建立了这些值函数的基本性质，并推导出相关的贝尔曼型（祖博夫型）函数方程。基于这一表征，我们发展了一种物理信息神经网络（NN）框架，通过将所推导的函数方程直接嵌入训练过程来学习值函数。通过两个数值示例验证了所提方法的有效性，展示了其精确估计DOS以及从学习到的值函数合成镇定控制器的能力。