AI 中文总结
研究随机微分方程原点稳定性对小扰动的鲁棒性条件，通过两种方式表达鲁棒性，在一定假设下证明相关性质，还提出基于鲁棒性分析工具的随机积分反步法。
AI 中文摘要
我们研究随机微分方程原点稳定性对小扰动具有鲁棒性的条件。通过两种方式表达鲁棒性：一是在不超过依赖于状态且在原点消失但在其他地方为正的界的小参数扰动下保持随机稳定性；二是通过随机输入到状态稳定性（ISS），其允许处处有非零扰动。在假设存在标称系统随机稳定性的李雅普诺夫函数下证明了前一种性质。在相同假设下，在合适的依赖于状态的扰动缩放条件下随机ISS成立。在比例有界扰动下保持随机指数稳定性，且即使没有扰动缩放也意味着指数ISS。最后，我们提出了一种新颖的纯反馈形式的随机积分反步法，该方法使用了我们鲁棒性分析的工具。
英文摘要
We study conditions under which stability of the origin of stochastic differential equations is robust to small perturbations. We express robustness in two ways, firstly in the sense that stochastic stability is maintained under small parametric perturbations not exceeding a state-dependent bound vanishing at the origin but positive elsewhere, and secondly via stochastic input-to-state stability (ISS) which allows non-zero perturbations everywhere. We prove the former property assuming the existence of a Lyapunov function certifying stochastic stability of the nominal system. Under the same assumption, stochastic ISS holds under a suitable state-dependent perturbation scaling. Stochastic exponential stability is maintained under proportionally bounded perturbations and implies exponential ISS even without perturbation scaling. Finally, we propose a novel approach to stochastic integrator backstepping in pure-feedback form that uses the tools from our robustness analysis.
CommentsSubmitted to IEEE Transactions on Automatic Control, 8 pages
