论文标题
在参数变化下，吸引区域边界的Hausdorff连续性，并应用于干扰恢复
Hausdorff Continuity of Region of Attraction Boundary Under Parameter Variation with Application to Disturbance Recovery
论文作者
论文摘要
考虑欧几里得空间或紧凑的riemannian歧管上的参数依赖性向量场。假设它具有依赖参数的初始条件和参数依赖性稳定双曲线平衡点。确定参数值集很有价值，我们称之为恢复集，其相应的初始条件位于相应稳定平衡点的吸引区域内。边界参数值是一个参数值，其相应的初始条件位于相应稳定平衡点吸引区域的边界。先前的算法通过计算边界参数值来估计其边界来估计恢复设置。这项工作的主要目的是为这些算法提供一类参数依赖的向量字段的理论理由。这包括证明，对于这些矢量字段，恢复集的边界由边界参数值组成，并且将满足算法所利用的属性，以计算这些所需的边界参数。这些证据所依赖的主要技术结果是，就此类别的向量字段而言，吸引区域边界的参数值的较小变化在适当的意义上持续变化。因此，这项工作的大多数致力于证明这一结果，这可能具有独立的利益。连续性证明是通过证明，对于这类矢量场，吸引区域允许将其包含的平衡点和周期轨道的稳定流形的分解分解为稳定的歧管，并且这种分解持续在对向量场的小扰动下。
Consider a parameter dependent vector field on either Euclidean space or a compact Riemannian manifold. Suppose that it possesses a parameter dependent initial condition and a parameter dependent stable hyperbolic equilibrium point. It is valuable to determine the set of parameter values, which we call the recovery set, whose corresponding initial conditions lie within the region of attraction of the corresponding stable equilibrium point. A boundary parameter value is a parameter value whose corresponding initial condition lies in the boundary of the region of attraction of the corresponding stable equilibrium point. Prior algorithms numerically estimated the recovery set by estimating its boundary via computation of boundary parameter values. The primary purpose of this work is to provide theoretical justification for those algorithms for a large class of parameter dependent vector fields. This includes proving that, for these vector fields, the boundary of the recovery set consists of boundary parameter values, and that the properties exploited by the algorithms to compute these desired boundary parameters will be satisfied. The main technical result which these proofs rely on is establishing that the region of attraction boundary varies continuously in an appropriate sense with respect to small variation in parameter value for this class of vector fields. Hence, the majority of this work is devoted to proving this result, which may be of independent interest. The proof of continuity proceeds by proving that, for this class of vector fields, the region of attraction permits a decomposition into a union of the stable manifolds of the equilibrium points and periodic orbits it contains, and this decomposition persists under small perturbations to the vector field.
