非线性年龄等级结构种群系统的边界控制问题

doi:10.12341/jssms22416

系统科学与数学 ›› 2023, Vol. 43 ›› Issue (2): 295-309.DOI: 10.12341/jssms22416

非线性年龄等级结构种群系统的边界控制问题

窦艺萌,何泽荣

杭州电子科技大学运筹与控制研究所, 杭州 310018

收稿日期:2022-06-05 修回日期:2022-09-17 出版日期:2023-02-25 发布日期:2023-03-16
通讯作者: 何泽荣,Email:zrhe@hdu.edu.cn
基金资助:
国家自然科学基金(11871185)资助课题.

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窦艺萌,何泽荣. 非线性年龄等级结构种群系统的边界控制问题[J]. 系统科学与数学, 2023, 43(2): 295-309.

DOU Yimeng, HE Zerong. Boundary Control Problems for a Nonlinear Hierarchical Age-Structured Population System[J]. Journal of Systems Science and Mathematical Sciences, 2023, 43(2): 295-309.

Boundary Control Problems for a Nonlinear Hierarchical Age-Structured Population System

DOU Yimeng, HE Zerong

Institute of Operational Research and Cybernetics, Hangzhou Dianzi University, Hangzhou 310018

Received:2022-06-05 Revised:2022-09-17 Online:2023-02-25 Published:2023-03-16

探究一类具有年龄等级结构的种群模型的边界调控问题,包括系统的能控性和最优控制.应用特征线方法和,Gronwall,不等式建立状态分布关于控制变量的连续依赖性;基于线性系统的能控性与集值映射不动点方法获得了非线性种群系统的近似能控性;运用,Ekeland,变分原理证明了最优策略的存在唯一性,构造适当的共轭系统和法向量对最优策略作出精细刻画.数值模拟结果展示了控制策略的可行性.

关键词： 个体等级, 边界控制, Ky Fan-Glicksberg不动点定理, 近似能控性, 最优控制, Ekeland变分原理

This article explores some boundary control problems for a hierarchical age-structured population model, which is of an integro-partial differential equation with a global feedback boundary condition; the control variable stands for input of baby individuals of the population. Firstly, the continuity of the population densities with respect to the control is established by characteristics method and Gronwall’s inequality. Then we prove that the population system is approximately controllable by means of a controllability result for linear systems and Ky Fan-Glicksberg fixed-point theorem for set-valued mappings. Thirdly, Ekeland’s variational principle is used to show the existence of unique optimal strategy, and finally, the optimal strategy is carefully depicted via an adjoint system and a normal vector. Moreover, some numerical experiments display the feasibility of the boundary control policy.

Key words: Hierarchy of individuals, boundary control, Ky Fan-Glicksberg’s fixed points, approximate controllability, optimal control, Ekeland’s principle

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