This paper studies a periodic optimal control problem governed by a one-dimensional system, linear with respect to the control $ u $, under an integral constraint on $ u $. We give conditions for which the value of the cost function at steady state with a constant control $ \bar u $ can be improved by considering periodic control $ u $ with average value equal to $ \bar u $. This leads to the so-called "over-yielding" met in several applications. With the use of the Pontryagin Maximum Principle, we provide the optimal synthesis of periodic strategies under the integral constraint. The results are illustrated on a single population model in order to study the effect of periodic inputs on the utility of the stock of resource.

中文翻译：

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输入积分约束下标量动力学的最优周期控制

本文研究了由一维系统控制的周期最优控制问题，该问题相对于控制$ u $是线性的，并且受到$ u $的积分约束。我们给出条件，可以通过考虑均值等于$ \ bar u $的周期控制$ u $来改善具有恒定控制$ \ bar u $的稳态下成本函数的值。这导致在几个应用程序中遇到所谓的“过度生产”。利用庞特里亚金最大原理，我们在积分约束下提供了周期策略的最佳综合。在单个人口模型上说明了结果，以便研究定期投入对资源存量效用的影响。