﻿ 环境污染下一类具有尺度结构种群系统的最优控制 Optimal Control of a Size-Structured System in a Polluted Environment

Vol.05 No.03(2016), Article ID:18307,7 pages
10.12677/AAM.2016.53044

Optimal Control of a Size-Structured System in a Polluted Environment

Jiangbi Liu1, Genquan Li1,2

1School of Mathematical Science, Lanzhou Jiaotong University, Lanzhou Gansu

2Zhuanglang No. 2 Middle School, Pingliang Gansu

Received: Jul. 21st, 2016; accepted: Aug. 14th, 2016; published: Aug. 17th, 2016

ABSTRACT

In this paper, we investigate the optimal harvesting for a class of size-structured population system in a polluted environment, making the maximum revenue by controlling the species harvest and inputting rates of the external toxin into the environment. Fixed point theory is used to obtain the existence and uniqueness of solution of the system. Optimality conditions are derived by means of tangent-normal cones and the technique of adjoint system. Some results in references are extended.

Keywords:Optimal Control, Size-Structure, Environment Pollution, Fixed Point Theory

1兰州交通大学数理学院，甘肃 兰州

2庄浪县第二中学，甘肃 平凉

1. 引言

(OH)

(1.1)

(H1)为常数。

(H2)

(H3)为常数。

(H4)

(H5)均为正常数，其中

(1.2)

，(1.3)

， (1.4)

1)

2)

。当时，则有

(1.5)

(1.6)

(1.7)

2. 最优性条件

(2.1)

， (2.2)

， (2.3)

(2.4)

， (2.5)

Optimal Control of a Size-Structured System in a Polluted Environment[J]. 应用数学进展, 2016, 05(03): 360-366. http://dx.doi.org/10.12677/AAM.2016.53044

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