Incremental Input-to-State Stability and Equilibrium Tracking for Stochastic Contracting Dynamics

TL;DR

This paper establishes conditions under which stochastic differential equations exhibit incremental input-to-state stability, enabling error estimation and equilibrium tracking in noisy environments, with applications to Fokker-Planck equations and various noise models.

Contribution

It introduces new stability criteria for stochastic contracting systems with deterministic inputs, extending to Fokker-Planck equations and different noise processes.

Findings

SDEs are incrementally noise- and input-to-state stable under contraction and Lipschitz conditions.

Error bounds for equilibrium tracking are derived in noisy stochastic environments.

Incremental stability is established for Fokker-Planck equations using Wasserstein metrics.

Abstract

In this paper, we study the contractivity of nonlinear stochastic differential equations (SDEs) driven by deterministic inputs and Brownian motions. Given a weighted $\ell_2$-norm for the state space, we show that an SDE is incrementally noise- and input-to-state stable if its vector field is uniformly contracting in the state and uniformly Lipschitz in the input. This result is applied to error estimation for time-varying equilibrium tracking in the presence of noise affecting both the system dynamics and the input signals. We consider both Ornstein-Uhlenbeck processes modeling unbounded noise and Jacobi diffusion processes modeling bounded noise. Finally, we turn our attention to the associated Fokker-Planck equation of an SDE. For this context, we prove incremental input-to-state stability with respect to an arbitrary $p$-Wasserstein metric when the drift vector field is uniformly contracting in the state and uniformly Lipschitz in the input with respect to an arbitrary norm.