We consider a simple crop irrigation model and study the optimal control which consists of maximizing the biomass production at harvesting time. A specificity of this work is to impose a quota on the water used for irrigation, in a context of limited resources. The model is written as a 2d non-autonomous dynamical system with a state constraint, and a non-smooth right member given by threshold-based soil and crop water stress functions. We show that when the water quota is below the threshold giving the largest possible production, the optimal strategy consists of irrigating once. We then show that the optimal solution can have one or several singular arcs, and therefore be better than simple bang-bang controls, as commonly used. The gains over the best bang-bang controls are illustrated on numerical simulations. These new feedback controls that we obtain are a promising first step towards the concrete application of control theory to the problem of optimal irrigation scheduling under water scarcity.

中文翻译：

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缺水条件下作物灌溉模型的优化控制

我们考虑了一个简单的作物灌溉模型，并研究了包括在收获时最大化生物量产量的最优控制。这项工作的一个特点是在资源有限的情况下对灌溉用水实行配额。该模型被写成具有状态约束的二维非自治动力系统，以及由基于阈值的土壤和作物水分胁迫函数给出的非光滑右成员。我们表明，当用水配额低于提供最大可能产量的阈值时，最佳策略包括灌溉一次。然后我们证明了最优解可以有一个或几个奇异的弧，因此比常用的简单的 bang-bang 控制更好。数值模拟说明了优于最佳 bang-bang 控件的收益。