Robust Finite‐Time Stability in Stochastic Systems via Homogeneous Approximation

This study extends the concept of homogeneous approximation to address homogeneous stochastic systems both at the origin and at infinity, capturing the bi‐limit behavior. We develop robust stability conditions that ensure stability for systems admitting such homogeneous approximations, even under stochastic influences. Building on this theoretical framework, we focus on the robustness of finite‐time stability in homogeneous stochastic nonlinear dynamical systems, particularly when subjected to sufficiently small affine inputs. Furthermore, we propose a robust finite‐time controller tailored for a class of affine systems, alongside a method to construct the required Lyapunov function. Specific conditions for the system's structure are provided to guarantee robust finite‐time stability. An example and simulation illustrate the effectiveness of the proposed approach, supporting its potential for advancing control strategies in complex, uncertain environments.