研讨一类基于年龄等级结构和空间扩散行为的种群系统模型的最优控制问题, 其状态系统由二阶非线性偏微分积分方程描述, 控制变量为种群初始分布, 指标泛函表征任意固定时间段后的种群资源净收益. 运用法锥理论并构造恰当的共轭系统方程建立了最大值原理, 扩充指标泛函且应用Ekeland变分原理及不动点方法证明了最优解的存在唯一性. 数值实验显示了所得理论成果的可应用性.
Abstract
This paper is concerned with an optimal control problem, in which the state system is described by a nonlinear second-order partial integro-differential equation, the control function stands for initial population distribution, and the index gives the net benefits from the population resources after an arbitrarily fixed time. Based on some conditions on model parameters, we establish the necessary optimality conditions (i.e., maximum principle), which provide the structure of optimal strategies and are given by a truncating function of the adjoint variable. Furthermore, we show that there is one and only one optimal control policy by the means of Ekeland's variational principle and fixed points reasoning of contraction mappings. Numerical experiments display the applicability of the obtained theoretical results.
关键词
随机游动 /
等级差异 /
初始分布 /
最优控制 /
法锥 /
变分原理
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Key words
Random walks /
hierarchy of ages /
initial distributions /
optimal controls /
normal cones /
variational principle
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中图分类号：
92B05
49K20
49J20
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参考文献
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脚注
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基金
国家自然科学基金(11871185)资助课题.
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