Background: Several countries nowadays and in the past face a birth rates below replacement level. To what extent should the fertility of this shrinking population be increased during a given planning period such that it approaches stationarity at the end as close as possible? Both immediate adaptation to the replacement level as well as delaying it to the end of the planning period are suboptimal. Methods: Distributed parameter optimal control theory provides an appropriate tool to ascertain the efficient intertemporal trade-off between costly birth control and zero population growth. Results: It turns out that the optimal adaptation rate of the net reproduction rate (NRR) balances between unacceptable adjustment costs for fertility and huge deviations of the terminal age composition from the desired stationary one. The optimal adaptation rate is monotonically increasing with a curvature that depends on the growth rates of the NRR, the fertile population, and the value of newborns. Contribution: The paper analytically characterizes the shape of the transition to a stationary population in an optimal way.

中文翻译：

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集中活力率向固定种群的最佳过渡（作者：Gustav Feichtinger、Stefan Wrzaczek）

背景：现在和过去有几个国家面临着低于更替水平的出生率。在给定的规划期内，这个不断减少的人口的生育率应该提高到什么程度，以便最终尽可能接近平稳？立即适应更换水平以及推迟到规划期结束都不是最佳选择。方法：分布式参数最优控制理论提供了一个适当的工具来确定昂贵的生育控制和零人口增长之间的有效跨期权衡。结果：事实证明，净生育率（NRR）的最佳适应率在不可接受的生育率调整成本和最终年龄构成与理想稳定年龄构成的巨大偏差之间取得了平衡。最佳适应率随着曲率单调递增，该曲率取决于 NRR 的增长率、可生育人口和新生儿的价值。贡献：本文以最佳方式分析描述了向固定种群过渡的形状。