非凸域与紧致流形中的噪声诱导导航
Noise-Induced Navigation in Non-convex Domains and Compact Manifolds
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中文总结 AI 辅助
该研究针对非凸域与紧致流形的导航问题，通过引入控制通道噪声构造初等反馈律，实现目标平衡点全局渐近稳定，数值实验验证了方法在带障碍欧几里得域及二维球面的有效性。
中文摘要 AI 辅助
本文研究设计反馈律以全局引导系统到达指定目标构型的问题。即使系统完全致动，拓扑障碍通常会导致全局渐近稳定的连续反馈律不存在。我们在随机环境中重新探讨该问题，允许噪声通过控制通道引入。结合局部李雅普诺夫稳定性与正回归的全局渐近稳定性准则，我们构造性地证明，对于带障碍的连通欧几里得域及无边界流形，可构造初等反馈律实现目标平衡点的全局渐近稳定。对于带障碍的欧几里得域，我们还表明该方法可自然扩展至求解带非凸约束的强凸函数极小值问题。数值实验验证了该方法在带圆形障碍的欧几里得域及二维球面上的有效性。此外，我们研究了存在非凸障碍时噪声强度的作用，此时系统可能表现出亚稳态。
英文摘要
In this note, we study the problem of designing a feedback law that globally steers a system to a prescribed target configuration. Even if the system is fully actuated, topological obstructions generally prevent the existence of globally asymptotically stabilizing continuous feedback laws. We revisit this problem in a stochastic setting by allowing noise to enter through the control channels. Using a criterion for asymptotic stability in the large that combines local Lyapunov stability with positive recurrence, we constructively show that one can construct elementary feedback laws that achieve global asymptotic stability in the large, of the target equilibrium point in connected Euclidean domains with obstacles and manifolds without boundary. For Euclidean domains with obstacles, we also show that the method extends naturally to the problem of finding the minimizer of a strongly convex function with non-convex constraints. Numerical experiments illustrate the effectiveness of the approach for Euclidean domains with circular obstacles and the two dimensional sphere. Additionally, we study the role noise strength when there is a non-convex obstacle, in which case the system might show metastability.
