Optimal Control Of Several Classes Of Stochastic Population Systems With Fractional Brownian Motion In A Polluted Environment

Since the 21 st century,the rapid development of science and technology has promoted the way of production and life of human beings,and it has become normal for human beings to obtain huge economic benefits.Due to the unilateral development of productive forces and excessive interference with nature,human beings have exceeded the carrying capacity of the ecological environment,resulting in the ecological environment crisis and worsening environmental pollution,especially the excessive discharge of various pollutants into the environment.At the same time,the biomass exchange materials and energy with the inorganic environment on which it depends for its survival,which leads to the increasing deposition of endotoxin in the organism,which seriously threatens the survival of human beings and other organisms.In view of the above analysis,it is necessary to establish the corresponding biological mathematical model and to analyze the survival of the population under the toxic effect of environmental pollutants.Secondly,Fractional Brownian motion,as a stochastic perturbation model in complex dynamical systems,has attracted close attention from scholars at home and abroad.At present,the control theory of stochastic population systems driven by fractional Brownian motion is still an open problem to be solved.Therefore,in this paper,the optimal control problems of several kinds of stochastic population systems with fractional Brownian motion in polluted environment are discussed.This paper consists of five chapters,and the research contents are as follows:The first chapter is the introduction.It firstly summarizes the research background and practical significance of this paper,then summarizes the development status and research results of biological population system at home and abroad,and then introduces the main content of this paper.The second chapter is the preliminary knowledge,which mainly lists some definitions,concepts,related lemmas and theorems needed in the process of proof.The third chapter mainly studies the stochastic single-population model which depends on fractional Brownian motion and age structure in polluted environment from two aspects:Firstly,a population model based on age structure and random disturbance in polluted environment is established.By using It?’s formula,Gronwall lemma and Fatou’s lemma,the existence of optimal control of the model is proved.Secondly,the paper presents a type of pollution environment depends on the fractional Brownian motion and spatial diffusion random single population system,considering the influence of the external environment of population spatial diffusion noise,using Galerkin method of the finite dimensional approximate thought,with the aid of It?’s formula,the stochastic differential equation theory and the Bellman-Gronwall-Type lemma,the existence and uniqueness of the solution of stochastic system.The fourth chapter,a class of nonlinear stochastic two-population model with fractional Brownian motion in polluted environment is presented,and the necessary conditions to optimize the performance of the control system are studied.Firstly,the existence and uniqueness of the solution of the system are proved by using the comparison principle and Banach compression mapping theory.Then,Hamiltonian function is constructed,and the existence of the optimal control is proved by using It?’s formula,stochastic differential equation related theory and maximum principle.The necessary conditions for the performance index of the harvest control system to reach the optimal are obtained.Under this condition,it is beneficial to maximize the overall economic benefit.The fifth chapter summarizes the main tasks and researches conclusions of this paper,and looks forward to future research and discussion directions.