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

Yu Kawano, Simone Betteti, Alexander Davydov, Francesco Bullo

TL;DR

This work develops a contraction-based framework for stochastic differential equations with deterministic inputs and additive noise, establishing incremental noise- and input-to-state stability (NISS) under uniform state contraction and input Lipschitzness. It then derives explicit stochastic equilibrium-tracking bounds for three noise scenarios using Ornstein–Uhlenbeck (unbounded) and Jacobi-diffusion (bounded) processes, covering deterministic and stochastic inputs and equilibria. Additionally, the paper extends these ideas to Wasserstein contractivity for the Fokker–Planck equation, proving $p$-Wasserstein incremental stability under the same structural assumptions. Collectively, the results provide quantitative, computable guarantees for stochastic solvers of time-varying convex problems and offer insights into the trade-offs between noise intensity, input variation, and contraction rates with practical implications for robust optimization and control under uncertainty.

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.

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

TL;DR

This work develops a contraction-based framework for stochastic differential equations with deterministic inputs and additive noise, establishing incremental noise- and input-to-state stability (NISS) under uniform state contraction and input Lipschitzness. It then derives explicit stochastic equilibrium-tracking bounds for three noise scenarios using Ornstein–Uhlenbeck (unbounded) and Jacobi-diffusion (bounded) processes, covering deterministic and stochastic inputs and equilibria. Additionally, the paper extends these ideas to Wasserstein contractivity for the Fokker–Planck equation, proving -Wasserstein incremental stability under the same structural assumptions. Collectively, the results provide quantitative, computable guarantees for stochastic solvers of time-varying convex problems and offer insights into the trade-offs between noise intensity, input variation, and contraction rates with practical implications for robust optimization and control under uncertainty.

Abstract

In this paper, we study the contractivity of nonlinear stochastic differential equations (SDEs) driven by deterministic inputs and Brownian motions. Given a weighted -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 -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.
Paper Structure (23 sections, 15 theorems, 91 equations)

This paper contains 23 sections, 15 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Input-to-State Stability of Deterministic Systems and Equilibrium Tracking
  5. Stochastic Differential Equations
  6. Ornstein-Uhlenbeck Process
  7. Jacobi Diffusion Process
  8. Incremental Noise- and Input-to-State Stability
  9. Stochastic Equilibrium Tracking
  10. Driven by Ornstein-Uhlenbeck Processes
  11. Deterministic Input, Tracking a Deterministic Curve
  12. Stochastic Input, Tracking a Deterministic Curve
  13. Stochastic Input, Tracking a Stochastic Curve
  14. Driven by Jacobi Diffusion Processes
  15. Stochastic Input, Tracking a Deterministic Curve
  16. ...and 8 more sections

Key Result

Proposition 1

Given an input-dependent vector field $F:\mathbb{R}^{n} \times \mathcal{U} \to \mathbb{R}^{n}$ and deterministic measurable inputs $u^{x},u^{y}:\mathbb{R}\to\mathcal{U}$, where $\mathcal{U} \subset \mathbb{R}^m$ is compact, consider a pair of systems: Assume that there exists a matrix $P=P^{\top}\succ \mathbb{0}_{n \times n}$ such that Then, for any finite $t \ge 0$, the ODE eq:ODE is incrementa

Theorems & Definitions (33)

  • Proposition 1: Incremental input-to-state stability
  • Proposition 2: Equilibrium Tracking
  • Proposition 4: Existence, uniqueness, and joint measurability of strong solutions
  • Definition 5: Infinitesimal generator
  • Proposition 6: Dynkin’s formula
  • Theorem 7: Incremental noise- and input-to-state stability
  • proof
  • Remark 1: Comparisons
  • Remark 2: Comparisons
  • Theorem 8: Stochastic Equilibrium Tracking: Deterministic Input, Tracking a Deterministic Curve
  • ...and 23 more