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Small-gain theorems for nonlinear stochastic systems with inputs and outputs II: Multiplicative white noise case

This paper is a continuation of the paper \cite{JL}, which focuses on exploring the global stability of nonlinear stochastic feedback systems on the nonnegative orthant driven by multiplicative white noise and presenting a couple of small-gain results. We investigate the dynamical behavior of pull-back trajectories for stochastic control systems and prove that there exists a unique globally attracting positive random equilibrium for those systems whose output functions either possess bounded derivatives or are uniformly bounded away from zero. In the first case, we first prove the joint measurability of both the pull-back trajectories and the metric dynamical system $θ$ with respect to the product $σ$-algebra $\mathscr{B}(\mathbb{R_+})\otimes\mathscr{F}_-$ and $\mathscr{B}(\mathbb{R_-})\otimes\mathscr{F}_-$, respectively, where $\mathscr{F}_-=σ\{ω\mapsto W_t(ω):t\leq0\}$ is the past $σ$-algebra and $W_t(ω)$ is an $\mathbb{R}^d$-valued two-sided Wiener process, and then combine the $\mathcal{L}^1$-integrability of the tempered random variable coming from the definition of the top Lyapunov exponent and the independence between the past $σ$-algebra and the future $σ$-algebra $\mathscr{F}_+=σ\{ω\mapsto W_t(ω):t\geq0\}$ to obtain the small-gain theorem by constructing the contraction mapping on an $\mathscr{F}_-$-measurable, $\mathcal{L}^1$-integrable and complete metric input space; in the second case, the sublinearity of output functions and the part metric play the main roles in the existence and uniqueness of globally attracting positive fixed point in the part of a normal, solid cone. Our results can be applied to well-known stochastic Goodwin negative feedback system, Othmer-Tyson positive feedback system and Griffith positive feedback system as well as other stochastic cooperative, competitive and predator-prey systems.