年龄相关的种群空间扩散系统的广义解与收获控制

Generalized solution and optimal harvesting control for age-dependent population spatial diffusion system

中文关键词  年龄相关的种群空间扩散系统  积分偏微分方程的广义解  最优收获控制  最优控制的必要条件  最优性组

英文关键词  age-dependent population spatial diffusion system  generalized solution for integral partial differential equations  optimal harvesting control  necessary conditions of optimal control  optimality system

中文摘要

      研究了由积分偏微分方程描述的年龄相关的种群空间扩散系统的收获控制问题.首先利用不动点方法证明了对于有界死亡率μ的系统广义解的存在性,但这是预备的结果.进一步,运用上述结果、先验估计和紧性定理,证明了对于在r=A附近无界的μ的系统解的存在惟一性.其次,利用类似方法得到系统最优收获控制的存在性.最后,利用Ga^teax微分和Lions的变分不等式理论,推得了控制为最优的必要条件;从而得到了由积分偏微分方程和变分不等式构成的最优性组.最优性组能够确定最优控制.还建立了表征最优控制的Euler-Lagrange组.这些结果可为种群系统控制问题的实际研究作为理论参考.

英文摘要

      The optimal harvesting control problem for the age-dependent population spatial diffusion system represented by integral partial differential equations is discussed.First,using a fixed-point method,the existence of the generalized solution for the system is proved for the bounded relative mortality μ;yet it is a preliminary result.Further,by utilising the preceeding result,prior estimates and compactness theorem,the existence and uniqueness of the generalized solution for the system is proved for unbounded μ near r=A.Next,by using analogous methods,the existence of the optimal harvesting control for the system is obtained. Finally,the necessary condition for a control to be optimal is deduced,using Ga^teax differentiation and Lions's theory of variational inequalities;and then the optimal system consisting of integral partial differential equations and variational inequalities is obtained.The optimal system can determine optimal controls.An Euler-Lagrange system characterized optimal control is also established.These results may serve as theoretical reference for practical researches of the control problem in population systems.