New robustness proof for stochastic differential equations via Lyapunov methods

A new paper by Moldenhauer, Nešić, Granzotto, Postoyan, and Teel (submitted to IEEE Transactions on Automatic Control) tackles a fundamental question in control theory: when does stability of a stochastic differential equation survive small disturbances? The authors show that if a Lyapunov function certifies stability of the nominal (undisturbed) system, then two forms of robustness follow. First, stability is preserved under parametric perturbations that vanish at the origin but are positive elsewhere—a state-dependent bound that keeps the system close to equilibrium. Second, stochastic input-to-state stability (ISS) holds, allowing non-zero perturbations everywhere if they are scaled appropriately by a function of the state. For exponential stability, proportional bounded perturbations (i.e., scaled by a constant) are sufficient, and this directly implies exponential ISS without extra scaling.

The paper also contributes a practical design tool: a novel stochastic integrator backstepping method for systems in pure-feedback form. Backstepping is a recursive control design technique, and this work extends it to stochastic settings using the robustness analysis framework. By leveraging Lyapunov functions and perturbation bounds, engineers can now construct controllers that guarantee stochastic stability even when their models are slightly wrong. This bridges a gap between deterministic robust control (ISS) and stochastic systems, with applications in robotics, autonomous vehicles, and any domain where random disturbances are inevitable. The 8-page paper, available on arXiv (2607.09127), provides rigorous proofs and sets the stage for more practical implementations.