In the present paper we propose reduced order methods as a reliable strategy to efficiently solve parametrized optimal control problems governed by shallow waters equations in a solution tracking setting. The physical parametrized model we deal with is nonlinear and time dependent: this leads to very time consuming simulations which can be unbearable, e.g., in a marine environmental monitoring plan application. Our aim is to show how reduced order modelling could help in studying different configurations and phenomena in a fast way. After building the optimality system, we rely on a POD-Galerkin reduction in order to solve the optimal control problem in a low dimensional reduced space. The presented theoretical framework is actually suited to general nonlinear time dependent optimal control problems. The proposed methodology is finally tested with a numerical experiment: the reduced optimal control problem governed by shallow waters equations reproduces the desired velocity and height profiles faster than the standard model, still remaining accurate.

中文翻译：

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参数化非线性时变最优流量控制的 POD-Galerkin 模型降阶：在浅水方程中的应用

在本文中，我们提出了降阶方法作为一种可靠的策略，以有效地解决解决方案跟踪设置中由浅水方程控制的参数化最优控制问题。我们处理的物理参数化模型是非线性的和时间相关的：这会导致非常耗时的模拟，这可能是无法忍受的，例如，在海洋环境监测计划应用程序中。我们的目标是展示降阶建模如何帮助快速研究不同的配置和现象。在构建最优系统后，我们依靠 POD-Galerkin 约简来解决低维约简空间中的最优控制问题。所提出的理论框架实际上适用于一般非线性时间相关的最优控制问题。