Study On Optimal Control Of Biased Population

At present,it is the blind pursuit of economic benefits and the development and the increasing utilization of population resources in the nature that result in the extinction of many populations and lead to the imbalance of the ecosystem.Therefore,it is of great significance to study the dynamical behavior and control of population system.On the one hand,the internal mechanism of population system can be explained by establishing mathematical model.On the other hand,it can effectively predict and take charge of the development and change of the population by establishing mathematical models and using different performance indicators.Nowadays,many scholars have done a lot of researches on the control of predator-prey system and competition system,and achieved many important results.However,there are few researches on the problem of biased population control.Therefore,it is necessary to study this kind of problems.The main research contents of this paper are as follows:The first chapter is the introduction.It mainly explains the research background and significance of this paper and describes the research status at home and abroad,in addition,it also lists the basic knowledges which is used in this paper.In the second chapter,we study the optimal tax strategy of biased population with biased level.Firstly,the stability condition of the positive equilibrium point of the system is obtained by using the stability theory of differential equation.Secondly,the optimal tax strategy is acquired by utilizing the extreme value principle of Pontryagin.In the third chapter,the optimal boundary control problem for biased population systems with scale-structure is studied.First,the conjugate system of the system is derived and the optimal boundary conditions are obtained by using the tangent cone-normal cone technique in nonlinear analysis.Second,Mazur's theorem in functional analysis is utilized to prove the existence of optimal boundary control.In the fourth chapter,we study the optimal control problem of biased population model with age structure in polluted environment.Firstly,the existence and uniqueness of the optimal control are proved by Banach fixed point theorem.Secondly,the necessary conditions of the optimal control are obtained by using the taper normal cone technique.Finally,the existence and uniqueness of the optimal control are proved by using the lower semicontinuousity and Ekeland's variational principle.