GLOBAL STABILITY OF FEEDBACK SYSTEMS WITH MULTIPLICATIVE NOISE ON THE NONNEGATIVE ORTHANT

摘要

We investigate the dynamical behavior of pull-back trajectories for feedback systems with multiplicative noise and prove that there exists a globally stable positive random equilibrium in the nonnegative orthant R-+(d), where the global stability means that all pull-back trajectories originating from nonnegative orthant converge to this positive random equilibrium almost surely. The output functions (feedback functions) are assumed to either possess bounded derivatives or be uniformly bounded away from zero. In the first case, we first prove the joint measurability of the metric dynamical system theta with respect to the sigma-algebra B(R_) circle times F_, where F_ = sigma{omega (sic) W-t(omega) : t = 0} is the past sigma-algebra and W-t(omega) is an R-d-valued two-sided Wiener process, and then combine the L-1-integrability of the tempered random variable coming from the definition of the top Lyapunov exponent and the independence between the past s-algebra and the future s-algebra F+ = sigma{omega (sic)W-t(omega) : t = 0} to obtain a globally stable random equilibrium by constructing the contraction mapping on an F_-measurable, L-1-integrable, and complete metric input space; in the second case, the sublinearity of output functions (feedback 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 a 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.