Optimal Control For Several Classes Of Age-dependent Population Systems In A Polluted Environment

With the development of social economy,people blindly pursue economic benefits while neglecting the protection of environmental resources,thus instigating the emergence of many environmental pollution problems.Such an environment is not conducive to the survival of biological populations.Therefore,it is of great realistic significance to study the impact of polluted environment on biological populations.In this paper,we study an optimal control for an age-dependent of hybrid system,competitive species with weighted and optimal con-trol for an age-dependent system with diffusion in a polluted environment.The existence and uniqueness of a nonnegative solution of this model is obtained by using a fixed point theorem.The existence of optimal control policy is carefully verified by using a maximum(minimum)principle,and the optimality conditions for the control problem are given by means of tangent-normal cone techniques.This paper mainly discusses the following sever-al aspects:(1)The model of optimal control for an age-dependent of population system with weight-ed in a polluted environment is established.The solution of the system is determined by the characteristic line method,and the existence uniqueness of the solution for the system is proved.Given performmance index,the existence of optimal control policy is verified by using the maximum principle and compactness,the necessary conditions for the control problem are given by applying the tangent-normal cone techniques.(2)The model of optimal control for an age-dependent of competitive population sys-tems with weighted in a polluted environment is established.The uniqueness of the solution of the system is given by using of Banach space theory,the necessary conditions for the opti-mal control are obtained by determining the conjugate system and using the tangent-normal cone techniques.(3)The model of the optimal control issue for age-structured with diffusion population system in a polluted environment is established.The existence of the solution of this system is given theorem,the existence of optimal control policy is verified by using the minimum principle and compactness,the necessary conditions for the control problem are given by applying the tangent-normal cone techniques.