A Small-Gain Theorem for Nonlinear Stochastic Systems with Inputs and Outputs I: Additive White Noise
A Small-Gain Theorem for Nonlinear Stochastic Systems with Inputs and Outputs I: Additive White Noise
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DOI：
10.1137/15m1044047
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发表时间：
2015-10
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影响因子：
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通讯作者：
Jifa Jiang;Xiang Lv
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文献类型：
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作者：
Jifa Jiang;Xiang Lv
本文研究了轨迹和平稳分布中由加性白噪声驱动的非线性随机方程的小增益定理。受 Marcondes de Freitas 和 Sontag \cite{FS3} 最近工作的启发，我们首先通过后向 Ito 积分在适当选择的输入空间中定义系统的 {\it "input-to-statecharacteristic算子"} $\mathcal{K}(u)$，然后对于给定的输出函数 $h$，将 $"gain\operator"$ 定义为输出函数 $h$ 和输入-to-state特征算子的组合输入空间上的$\mathcal{K}(u)$。假设输出函数在通常的向量阶上是保序或反保序的，并且输出函数的全局 Lipschitz 常数小于线性矩阵的负主特征值的绝对值。然后我们证明了所谓的“小增益定理”：增益算子有唯一的不动点，该不动点处的输入状态特征算子的图像是由随机系统生成的随机动力系统的全局吸引随机平衡。在同样假设输出函数的Lipschitz常数与稳定线性矩阵最大实部之间的关系的情况下，我们证明了随机系统具有唯一的平稳分布，将其视为小增益定理的平稳分布版本。这些结果可以应用于随机合作、竞争和捕食者-被捕食系统，甚至其他系统。
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 Ito 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.