General nonlinear stochastic systems motivated by chemostat models: Complete characterization of long-time behavior, optimal controls, and applications to wastewater treatment - Nguyen, Dang H.; Nguyen, Nhu N.; Yin, George (Stochastic processes and their applications, 2020)
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 λ. If λ≤0, the bacteria will die out, which means that the process does not operate. If λ>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 λ=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.