Optimal harvesting control for an age-dependent competing population with diffusion
© Fu et al.; licensee Springer 2014
Received: 23 January 2014
Accepted: 28 May 2014
Published: 23 September 2014
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.
MSC: 35B10, 49K05, 65L12, 92B05.
represents the profit due to harvesting.
Throughout this paper, we assume that:
(A1): , , where is a constant.
(A2): , a.e. in , , where , .
(A3): , , where is a positive constant (, ).
(A4): , , , , where is a constant, .
The rest of this paper is organized as follows. In Section 2, we prove that under the assumptions listed above, the system has a unique non-negative solution. In Section 3, the necessary conditions of optimality for the control problems is given. In the final section, we prove the existence and uniqueness of the optimal control.
2 The existence and uniqueness of solution for system (1.2)
For the sake of convenience, we introduce the following definitions of the solution.
where , .
, , , where is a positive constant;
is continuous in .
It is obvious that and is a contraction on , so there is a unique fixed point, which is the solution of system (1.2).
Inequality (2.9) implies that is continuous with respect to . This completes the proof. □
3 The necessary conditions
Before stating our main results, we prove the following lemma, which is useful in proving our results.
where , , .
where , , .
Consequently, the proof is completed. By the same argument as in (2.9), the following lemma holds. □
In order to give the necessary conditions, we introduce the adjoint equations of (3.1).
where, , , , , are constant and independent of, , , .
We are now able to state the main result in this section.
This completes the proof. □
4 The existence and uniqueness of optimal control
In this section, we prove the existence and uniqueness of optimal control. From , we give the following lemma.
The main conclusion is presented as follows:
Ifis small enough, then Problem (1.1) has a unique optimal control in.
we shall prove that is strictly monotone. Indeed, denote by and the state corresponding to controls and , respectively.
where is the solution of Problem (3.5) corresponding to the control ().
Hence, is strictly monotone. Consequently, is strictly concave in .
It is clear that is concave in . It follows that the nonlinear function is upper weakly semi-continuous. Since is a closed, convex, and bounded subset in and is strictly convex in , by Lemma 4.1, attains the unique maximum in , which implies Problem (1.1) has a unique optimal control in . □
The authors would like to thank the anonymous referees for their valuable comments on and suggestions regarding the original manuscript. The project is supported by Graduate Student Innovation of Jilin Normal University (201114) and the Department of Education of Jilin Province (2013445).
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