研究一类基于个体年龄等级结构的种群模型的最优收获问题, 状态系统由偏微分－积分方程描述, 目标泛函由状态变量与控制变量的双线性形式给出. 运用极值化序列和紧性推理确立了最优策略的存在性, 借助切锥法锥与共轭系统技巧对最优解做精确刻画. 所得结果为多种群控制及模型应用奠定基础.

We investigate an optimal harvesting problem for a class of hierarchical age-structured population model, in which the state system is presented by a partial integro-differential equation and the object functional is of bilinear form in state and control variables. The basic assumption is that all of the individuals survive in the internal environment, which is defined, for an individual of age $a$, as the sum of the number of individuals with age $a^{\prime }\le a$ and a weighted number with age $a^{\prime }>a$. The existence of the optimal policies is established by means of the truncated environment, a maximizing sequence and compactness reasoning. Furthermore, the characteristic description of the optimal controllers is carefully derived via tangent-normal cones and a well-posed adjoint system. The results obtained lay a solid basis for the approximation to the optimal strategies and model applications.