Analyzing nonlinear systems with attracting robust invariant sets (RISs) requires estimating their domains of attraction (DOAs). Despite extensive research, accurately characterizing DOAs for general nonlinear systems remains challenging due to both theoretical and computational limitations, particularly in the presence of uncertainties and state constraints. In this paper, we propose a novel framework for the accurate estimation of safe (state-constrained) and robust DOAs for discrete-time nonlinear uncertain systems with continuous dynamics, open safe sets, compact disturbance sets, and uniformly locally $\ell_p$-stable compact RISs. The notion of uniform $\ell_p$ stability is quite general and encompasses, as special cases, uniform exponential and polynomial stability. The DOAs are characterized via newly introduced value functions defined on metric spaces of compact sets. We establish their fundamental mathematical properties and derive the associated Bellman-type (Zubov-type) functional equations. Building on this characterization, we develop a physics-informed neural network (NN) framework to learn the corresponding value functions by embedding the derived Bellman-type equations directly into the training process. To obtain certifiable estimates of the safe robust DOAs from the learned neural approximations, we further introduce a verification procedure that leverages existing formal verification tools. The effectiveness and applicability of the proposed methodology are demonstrated through four numerical examples involving nonlinear uncertain systems subject to state constraints, and its performance is compared with existing methods from the literature.

翻译：分析具有吸引鲁棒不变集（RISs）的非线性系统需要估计其吸引域（DOAs）。尽管已有广泛研究，但由于理论和计算上的限制，特别是在存在不确定性和状态约束的情况下，准确表征一般非线性系统的吸引域仍然具有挑战性。本文针对具有连续动态、开放安全集、紧致扰动集以及一致局部 $\ell_p$ 稳定的紧致鲁棒不变集的离散时间非线性不确定系统，提出了一种用于精确估计安全（状态约束）且鲁棒的吸引域的新框架。一致 $\ell_p$ 稳定性的概念非常一般，包含了一致指数稳定性和多项式稳定性作为特例。吸引域通过新引入的、定义在紧致集度量空间上的值函数来表征。我们建立了这些值函数的基本数学性质，并推导了相关的贝尔曼型（祖博夫型）泛函方程。基于此表征，我们开发了一个物理信息神经网络（NN）框架，通过将推导出的贝尔曼型方程直接嵌入训练过程来学习相应的值函数。为了从学习到的神经近似中获得安全鲁棒吸引域的可验证估计，我们进一步引入了一个验证程序，该程序利用了现有的形式化验证工具。所提方法的有效性和适用性通过四个涉及受状态约束的非线性不确定系统的数值算例得到验证，并将其性能与文献中的现有方法进行了比较。