Abstract
This paper is mainly concerned with optimal harvesting policy for competing species with age dependence and diffusion. The existence and uniqueness of solutions for the system are proved by using the Banach fixed point theorem. The existence and uniqueness of optimal control are discussed by Ekeland’s variational principle. The maximum principle is obtained.
Highlights
1 Introduction The optimal harvesting control of age-structured and size-structured single species have been widely studied in the literature [ – ]
The control problems of multi-species have been investigated in the literature [ – ]
We introduce the following notations: C S; L ( ) = h : S → L ( ); h continuous, AC S; L ( ) = h : S → L ( ) : h(a + ·, t + ·) : (, α) → L ( )
Summary
The optimal harvesting control of age-structured and size-structured single species have been widely studied in the literature [ – ]. Our purpose is to consider the optimal control of the profit functional for diffusion population. We deal with the following optimal harvesting problem:. N, where pi(a, t, x) represents the density of the ith population. The function pi (a, x) gives the initial density distribution of the population, and ui(a, t, x) represents the harvesting effort function, which is the control variable in the model and satisfies n ui ∈ Ui = vi ∈ L (Q)| ≤ γi (a, t, x) ≤ vi(a, t, x) ≤ γi (a, t, x) a.e. in Q , U = Ui, i=. In Section , the necessary conditions of optimality for the control problems is given. We prove the existence and uniqueness of the optimal control.
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