On robustness, input-to-state stability and backstepping for stochastic differential equations

What the math gives to ML

The paper provides a constructive robustness mechanism for stochastic differential equations: a Lyapunov certificate for a nominal system can be converted into robustness against state-dependent perturbations and stochastic input-to-state stability. Its strongest transferable asset is the distinction between perturbations that vanish near the equilibrium and proportionally bounded perturbations, with stochastic exponential stability yielding exponential ISS without requiring vanishing perturbations. In neural-network training, this suggests monitoring a Lyapunov-like energy of parameter dynamics and scaling optimizer noise, gradient error, or parameter perturbations according to the local Lyapunov decay margin. The method makes a falsifiable prediction: instability begins when the estimated perturbation gain exceeds the nominal Lyapunov decay rate.