A SMALL-GAIN THEOREM FOR NONLINEAR STOCHASTIC SYSTEMS WITH INPUTS AND OUTPUTS I: ADDITIVE WHITE NOISE

作者：Jiang, Jifa; Lv, Xiang^*

来源：SIAM Journal on Control and Optimization, 2016, 54(5): 2383-2402.

DOI：10.1137/15M1044047

• stochastic control systems
• input and output
• small-gain theorem
• random dynamical systems
• stochastic equilibrium
• global stability

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 [SIAM J. Control Optim., 53 (2015), pp. 2657-2695], we first define the input-to-state characteristic operator K(u) of the system in a suitably chosen input space via a backward Ito integral, and then for a given output function h we define the gain operator as the composition of output function h and the input-to-state characteristic operator 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 the linear matrix. Then we prove the so-called small-gain theorem: the gain operator has a unique fixed point; the image for the 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 the maximal real part of the stable linear matrix, we prove that the stochastic system has a unique stationary distribution, which is regarded as a stationary distribution version of the small-gain theorem. These results can be applied to stochastic cooperative, competitive, and predator-prey systems, or even others.

• 出版日期2016
• 单位上海师范大学