General nonlinear stochastic systems motivated by chemostat models: Complete characterization of long-time behavior, optimal controls, and applications to wastewater treatment
General nonlinear stochastic systems motivated by chemostat models: Complete characterization of long-time behavior, optimal controls, and applications to wastewater treatment
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DOI：
10.1016/j.spa.2020.01.010
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发表时间：
2020-08-01
影响因子：
1.4
通讯作者：
Yin, George
中科院分区：
文献类型：
--
作者：
Nguyen, Dang H.;Nguyen, Nhu N.;Yin, George
受随机恒化器模型的启发，本文研究了一类非线性随机微分方程系统。在第一部分中，系统被配制成一个混合开关扩散。所考虑的系统的渐近行为的一个完整的表征。它示出的长期性能的系统可以通过使用一个实值参数λ进行分类。如果λ 0，系统有一个不变的概率测度，并且解过程的转移概率收敛到不变测度的转移概率。并给出了收敛速度。本文的一个显著特点是临界情况lambda = 0也被考虑。最后给出数值例子来说明我们的结果.在第二部分的文件中，控制扩散与长期平均目标函数进行处理。导出了相应的Hamilton-Jacobi-Bellman（HJB）方程，并证明了最优马尔可夫控制的存在性。本文的分析技术和方法可以应用于许多其他的随机Kolmogorov系统。(C)2020爱思唯尔B. V.保留所有权利。
This paper focuses on a general class of systems of nonlinear stochastic differential equations, inspired by stochastic chemostat models. In the first part, the system is formulated as a hybrid switching diffusion. A complete characterization of the asymptotic behavior of the system under consideration is provided. It is shown that the long-term properties of the system can be classified by using a real-valued parameter lambda. If lambda 0, the system has an invariant probability measure and the transition probability of the solution process converges to that of the invariant measure. The rate of convergence is also obtained. One of the distinct features of this paper is that the critical case lambda = 0 is also considered. Moreover, numerical examples are given to illustrate our results. In the second part of the paper, controlled diffusions with a long-run average objective function are treated. The associated Hamilton-Jacobi-Bellman (HJB) equation is derived and the existence of an optimal Markov control is established. The techniques and methods of analysis in this paper can be applied to many other stochastic Kolmogorov systems. (C) 2020 Elsevier B.V. All rights reserved.