Transient instability in nonlinear stochastic dynamical systems is a fundamental limitation in safety-critical aerospace applications, particularly during powered descent and landing where failure is driven by finite-time excursions rather than asymptotic divergence. Classical notions of mean-square or asymptotic stability are therefore insufficient for certification and design. This paper develops a logarithmic-norm-based framework for finite-time transient stability analysis of nonlinear Ito stochastic differential equations. The approach extends matrix measures to nonlinear mappings in a Lipschitz sense, enabling efficient characterization of instantaneous perturbation growth without local linearization. Using Ito calculus, bounds on the mean and variance of transient growth are derived, providing conditions for non-positive finite-time mean growth and probabilistic bounds on instability events. The analysis highlights a key distinction between mean and sample-path behavior, showing that stability in expectation does not guarantee pathwise finite-time safety, and that almost-sure transient stability cannot generally be ensured under stochastic diffusion. The framework is extended to data-constrained stochastic dynamics in navigation and estimation, revealing a trade-off between estimation consistency and transient robustness due to continuous data injection. Demonstrations with flight-like lunar lander telemetry show that similar mean trajectories can exhibit significantly different transient stability behaviour, and that mission failure correlates with accumulation of transient instability over short critical intervals. These results motivate probabilistic finite-time stability metrics for safety-critical autonomous systems.

翻译：非线性随机动力系统中的瞬态不稳定性是安全关键型航空航天领域的根本性限制，尤其在动力下降与着陆阶段，失效由有限时间偏离而非渐近发散驱动。因此，经典均方稳定性或渐近稳定性概念不足以支撑系统认证与设计。本文提出基于对数范数的框架，用于非线性伊藤随机微分方程的有限时间瞬态稳定性分析。该方法将矩阵测度以Lipschitz意义扩展到非线性映射，无需局部线性化即可高效表征瞬时扰动的增长。通过伊藤积分推导了瞬态增长均值与方差的有界性，给出了非正有限时间均值增长条件及不稳定事件的概率边界。分析揭示了均值行为与样本路径行为之间的关键差异：期望意义上的稳定性无法保证路径有限时间安全性，且在随机扩散条件下几乎必然瞬态稳定性通常无法实现。该框架进一步扩展至导航与估计中数据约束的随机动力学，揭示了连续数据注入导致的估计一致性与瞬态鲁棒性之间的权衡。基于类月面着陆器遥感的演示表明，相似均值轨迹可能呈现显著不同的瞬态稳定性行为，而任务失效与短关键窗口内瞬态不稳定性的累积相关。这些结果为安全关键型自主系统提供了概率有限时间稳定性指标的动机。