A small-gain theorem for nonlinear stochastic systems with inputs and outputs I: Additive white noise

TL;DR

This paper establishes a small-gain theorem for nonlinear stochastic systems with additive white noise, ensuring unique stochastic equilibria and stationary distributions under certain Lipschitz and eigenvalue conditions.

Contribution

It introduces an input-to-state characteristic operator and a gain operator framework for stochastic systems, extending small-gain results to systems driven by white noise.

Findings

Unique fixed point of the gain operator ensures a globally attracting stochastic equilibrium.

Existence of a unique stationary distribution for the stochastic system.

Applicable to various stochastic biological and cooperative systems.

Abstract

This paper studies a small-gain theorem for nonlinear stochastic equations driven by additive white noise in both trajectories and stationary distribution. Motivated by the most recent work of Marcondes de Freitas and Sontag \cite{FS3}, we firstly define the {\it "input-to-state characteristic operator"} $\mathcal{K}(u)$ of the system in a suitably chosen input space via backward It\^o integral, and then for a given output function $h$, define the $"gain\ operator"$ as the composition of output function $h$ and the input-to-state characteristic operator $\mathcal{K}(u)$ on the input space. Suppose that the output function is either order-preserving or anti-order-preserving in the usual vector order and the global Lipschitz constant of the output function is less than the absolute of the negative principal eigenvalue of linear matrix. Then we prove the so-called {\it "small-gain theorem"}: the gain operator has a unique fixed point, the image for input-to-state characteristic operator at the fixed point is a globally attracting stochastic equilibrium for the random dynamical system generated by the stochastic system. Under the same assumption for the relation between the Lipschitz constant of the output function and maximal real part of stable linear matrix, we prove that the stochastic system has a unique stationary distribution, which is regarded as a stationary distribution version of small-gain theorem. These results can be applied to stochastic cooperative, competitive and predator-prey systems, or even others.