A New Converse Lyapunov Theorem for Global Exponential Stability and Applications to Stochastic Approximation
A New Converse Lyapunov Theorem for Global Exponential Stability and Applications to Stochastic Approximation
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全局指数稳定性的新逆李雅普诺夫定理及其在随机逼近中的应用
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发表时间：
2022
期刊：
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通讯作者：
M. Vidyasagar
中科院分区：
文献类型：
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作者：
M. Vidyasagar
本文在适当的条件下，给出了随机逼近算法收敛的一个简单直观的证明。这里的主要结果可以与Borkar和Meyn(2000)基于常微分方程方法的结果相比较，即表明算法的样本路径收敛于相关常微分方程的确定性轨迹。与之相反，目前的证明是基于Gladyshev(1965)首次提出的鞅理论。因此，与以前的论文相比，这里的假设更少。证明的一个重要部分是全局指数稳定性的一个新的逆李雅普诺夫定理。除了它在随机逼近理论中的应用外，这个新的逆Lyapunov定理对稳定性理论的研究人员也是有用的。
In this paper, we give a simple and direct proof of the convergence of the stochastic approximation algorithm under suitable conditions. The main result here can be compared to that in Borkar and Meyn (2000), which is based on the ODE method, that is, showing that the sample paths of the algorithm converge towards the deterministic trajectories of an associated ODE. In contrast, the present proof is based on martingale theory, first proposed in Gladyshev (1965). Consequently, there are fewer assumptions here compared to previous papers. An important part of the proof is a new converse Lyapunov theorem for global exponential stability. Aside from its application to stochastic approximation theory, this new converse Lyapunov theorem would be useful for researchers in stability theory.