Inner-approximate reachability analysis involves calculating subsets of reachable sets, known as inner-approximations. This analysis is crucial in the fields of dynamic systems analysis and control theory as it provides a reliable estimation of the set of states that a system can reach from given initial states at a specific time instant. In this paper, we study the inner-approximate reachability analysis problem based on the set-boundary reachability method for systems modelled by ordinary differential equations, in which the computed inner-approximations are represented with zonotopes. The set-boundary reachability method computes an inner-approximation by excluding states reached from the initial set's boundary. The effectiveness of this method is highly dependent on the efficient extraction of the exact boundary of the initial set. To address this, we propose methods leveraging boundary and tiling matrices that can efficiently extract and refine the exact boundary of the initial set represented by zonotopes. Additionally, we enhance the exclusion strategy by contracting the outer-approximations in a flexible way, which allows for the computation of less conservative inner-approximations. To evaluate the proposed method, we compare it with state-of-the-art methods against a series of benchmarks. The numerical results demonstrate that our method is not only efficient but also accurate in computing inner-approximations.

翻译：内逼近可达性分析涉及计算可达集的子集，即内逼近。该分析在动态系统分析与控制理论领域中至关重要，因为它能够可靠地估计系统从给定初始状态在特定时刻可达的状态集合。本文研究基于集合边界可达性方法的常微分方程系统内逼近可达性分析问题，其中计算得到的内逼近用带状区域表示。集合边界可达性方法通过排除从初始集合边界出发可达的状态来计算内逼近。该方法的效果高度依赖于初始集合精确边界的有效提取。为此，我们提出利用边界矩阵与分块矩阵的方法，能够高效提取并细化由带状区域表示的初始集合的精确边界。此外，我们还通过灵活收缩外逼近来增强排除策略，从而能够计算保守性更低的内逼近。为了评估所提方法，我们将其与一系列基准问题上的最新方法进行对比。数值结果表明，我们的方法在计算内逼近时不仅高效，而且精确。