In this contribution we propose reduced order methods to fast and reliably solve parametrized optimal control problems governed by time dependent nonlinear partial differential equations. Our goal is to provide a general tool to deal with the time evolution of several nonlinear optimality systems in many-query context, where a system must be analysed for various physical and geometrical features. Optimal control is a tool which can be used in order to fill the gab between collected data and mathematical model and it is usually related to very time consuming activities: inverse problems, statistics, etc. Standard discretization techniques may lead to unbearable simulations for real applications. We aim at showing how reduced order modelling can solve this issue. We rely on a space-time POD-Galerkin reduction in order to solve the optimal control problem in a low dimensional reduced space in a fast way for several parametric instances. The generality of the proposed algorithm is validated with a numerical test based on environmental sciences: a reduced optimal control problem governed by Shallow Waters Equations parametrized not only in the physics features, but also in the geometrical ones. We will show how the reduced model can be useful in order to recover desired velocity and height profiles more rapidly with respect to the standard simulation, not loosing in accuracy.

中文翻译：

---

用于参数流控制问题和应用的降阶方法

在本文中，我们提出了降阶方法，以快速可靠地解决由时间相关的非线性偏微分方程控制的参数化最优控制问题。我们的目标是提供一个通用工具来处理许多查询环境中几个非线性最优系统的时间演化，其中必须分析系统的各种物理和几何特征。最佳控制是一种可用于填补收集的数据与数学模型之间的空白的工具，它通常与非常耗时的活动有关：逆问题，统计信息等。标准离散化技术可能会导致实际应用中难以忍受的模拟。我们旨在展示降阶建模如何解决此问题。我们依靠时空POD-Galerkin降阶，以便针对几个参数实例快速解决低维降维空间中的最优控制问题。该算法的通用性已通过基于环境科学的数值测试得到了验证：由浅水方程控制的减少的最优控制问题，不仅在物理特征上而且在几何特征上都进行了参数化。我们将展示简化的模型如何有用，以便相对于标准仿真更快地恢复所需的速度和高度轮廓，而不会降低精度。由浅水方程控制的最优控制问题的减少，不仅在物理特征中，而且在几何特征中。我们将展示简化的模型如何有用，以便相对于标准仿真更快地恢复所需的速度和高度轮廓，而不会降低精度。由浅水方程控制的最优控制问题的减少，不仅在物理特征中，而且在几何特征中。我们将展示简化的模型如何有用，以便相对于标准仿真更快地恢复所需的速度和高度轮廓，而不会降低精度。