具有空间扩散和尺度结构的非线性害鼠模型的最优不育控制  

Optimal contraception control for a nonlinear vermin model with spatial diffusion and size-structure

作　　者：张泰年 雒志学[2] 王汝军 ZHANG Tai-nian;LUO Zhi-xue;WANG Ru-jun(School of Environmental and Municipal Engineering,Lanzhou Jiaotong University,Lanzhou Gansu 730070,China;School of Mathematics and Physics,Lanzhou Jiaotong University,Lanzhou Gansu 730070,China;School of Mathematics and Statistics,Hexi University,Zhangye Gansu 734000,China)

机构地区：[1]兰州交通大学环境与市政工程学院,甘肃兰州730070 [2]兰州交通大学数理学院,甘肃兰州730070 [3]河西学院数学与统计学院,甘肃张掖734000

出　　处：《控制理论与应用》2023年第9期1555-1561,共7页Control Theory & Applications

基　　金：国家自然科学基金项目(11561041);甘肃省自然科学基金项目(23JRRG0006);河西学院校长基金创新团队项目(CXTD2023006)资助.

摘　　要：本文讨论了一类具有尺度结构的非线性时变害鼠扩散模型的适定性及最优不育控制问题.状态系统由二阶偏微分–积分方程描述,此系统有一种重要的特殊情形,即死亡率分为自然死亡率和额外死亡率,系统的解关于尺度和空间位置可分离,从而将系统分为两个子系统,利用比较原则和不动点定理证明了变量分离型解的存在唯一性和非负有界性.本文运用Mazur定理证明了最优策略的存在性,导出共轭系统并借助凸集的切锥–法锥技巧给出了最优策略的必要性条件,为模型的实际应用奠定了理论基础.最后,采用向后差分格式和追赶法分别对子系统的解进行了数值模拟.The paper investigates the well-posedness and optimal contraception control problem for a class of sizestructured nonlinear time-varying vermin diffusion model.The state system is described by a second-order partial integrodifferential equation.This system has an important particular situation,that is,the mortality rate is divided into natural mortality rate and additional mortality rate.The solution of the system is separable with respect to size and spatial position,thus dividing the system into two subsystems.The comparison principle and fixed point theorem are used to prove the existence,uniqueness,non-negativity and boundedness of the separable form of the solution.The existence of the optimal strategy is proved by Mazur’s theorem.The adjoint system is derived and the necessary conditions for the optimal strategy are given by means of tangent-normal cones technique of the convex set.The results lay a theoretical foundation for the practical applications of the model.Finally,the backward difference scheme and chasing method are used to simulate the solutions of the subsystems.

关 键 词：空间扩散 尺度结构 不育控制 可分离死亡率 有限差分法 

分 类 号：R184.35[医药卫生—流行病学] O232[医药卫生—公共卫生与预防医学]