The Optimal Scale Control For A Nonlinear Forest Population System With Periodic

This paper studies a nonlinear forest population system with periodic (P): here Q=(0,L)×(0,T),T is a Fixed time,p(l,t) is the population density of the forest in the time t and the size l, L is the maximum diameter for the population of the individuals; we have p(l,t)= 0,when l≥L,because of the practical significance of population, w(l) is The expected value function of the population increase in l,μ(l,t)is the natural martality of the population,μe(l,t;S) is the excess mortality of the population, presenting a decrease in population due to the deterioration of the living environment of the external or human cutting, β(l,t;S) is the reproduction rate of the population, S(t) is the scale population depending on u(l,t) in the time t. The weighted function u(l,t) is the control variables, f(l,t) is the Populations of the external disturbance function.In this paper, we have proved the existence and uniqueness of the generalized solutions for the system (P), by using the theory of partial differential equations. Finally, utilizing the compactness theorem and J.L.Lions theory for optimal control, we have proved the existence of the optimal control for the system (P). By means of variational inequality theory we have got the necessary conditions and optimality system for a control to be optimal.