Bounding Transient Instability in Sensor Data Injected Nonlinear Stochastic Flight Dynamics

Abstract: 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.