The Optimal Boundary Control Of A Population System With Size-structure And Weighted Size

In this paper,we discuss the optimal boundary control problem for a population system(P):of size-structure and weighted size.Referring to Q(28)(0,m)?(0,T),the fixed constant m is the largest scale and T is the control period of the population.State variable p(s,t)represents the density function of the population of s at the time scale.g(s)represents individual scale growth rate.N(t)and M(t)represent the weighted total population of t time.?(s,t,M(t))represents the birth rate of the individual scale s based on population weighted total amount of M(t).?(s,t,N(t))represents the death rate of the individual scale s based on population weighted total amount of N(t).f(s)represents the external migration of populations.?(x,t)represents a weight function to control the total population.?(x,t)is the female proportion of population density at t time.u(t)represents the control function of the new individual.In this paper,by using the theory of partial differential equations,the existence and uniqueness of the generalized solutions and the regularity of the system(P)are proved.By using Mazur's conclusion,the existence of optimal boundary control of the system(P)is proved.By means of Gateax's differential and Lion's variational inequality theory,we get the necessary condition and the optimal set of the optimal control.