Contraction analysis of nonlinear random dynamical systems
Contraction analysis of nonlinear random dynamical systems
复制标题
非线性随机动力系统的收缩分析
DOI：
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发表时间：
2013
期刊：
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通讯作者：
J. Slotine
中科院分区：
文献类型：
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作者：
Nicolas Tabareau;J. Slotine
为了将收缩分析引入到随机和贝叶斯系统的非常富有成效和热门的领域中，我们在这里将引用{Lohmiller 98}中描述的理论扩展到随机微分方程。我们提出了新的定义的收缩（几乎肯定收缩和收缩均方），它允许掌握随机系统的演化以两种方式。第一个保证最终指数收敛的系统几乎所有的提请，而另一个保证指数收敛到一个唯一的轨迹在$L_2$。然后，我们说明了这种扩展的相对简单性，通过分析常见的确定性属性存在的噪声。具体而言，我们分析了随机梯度下降，噪声对振荡器同步的影响和扩展的组合属性的合同系统的随机情况。这是将收缩系统的有趣和简化性质与概率方法相结合的第一步。
In order to bring contraction analysis into the very fruitful and topical fields of stochastic and Bayesian systems, we extend here the theory describes in cite{Lohmiller98} to random differential equations. We propose new definitions of contraction (almost sure contraction and contraction in mean square) which allow to master the evolution of a stochastic system in two manners. The first one guarantees eventual exponential convergence of the system for almost all draws, whereas the other guarantees the exponential convergence in $L_2$ of to a unique trajectory. We then illustrate the relative simplicity of this extension by analyzing usual deterministic properties in the presence of noise. Specifically, we analyze stochastic gradient descent, impact of noise on oscillators synchronization and extensions of combination properties of contracting systems to the stochastic case. This is a first step towards combining the interesting and simplifying properties of contracting systems with the probabilistic approach.