Concentration of Stochastic System Trajectories with Time-varying Contraction Conditions

We establish two concentration inequalities for nonlinear stochastic system under time-varying contraction conditions. The key to our approach is an energy function termed Averaged Moment Generating Function (AMGF). By combining it with incremental stability analysis, we develop a concentration inequality that bounds the deviation between the stochastic system state and its deterministic counterpart. As this inequality is restricted to single time instance, we further combine AMGF with martingale-based methods to derive a concentration inequality that bounds the fluctuation of the entire stochastic trajectory. Additionally, by synthesizing the two results, we significantly improve the trajectory-level concentration inequality for strongly contractive systems. Given the probability level $1-\delta$, the derived inequalities ensure an $\text{mO}(\sqrt{\log (1\mathrm{/}\delta ))}$ bound on the deviation of stochastic trajectories, which is tight under our assumptions. Our results are exemplified through a case study on stochastic safe control.