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.

翻译：在这一贡献中,我们提议减少订单方法,以快速和可靠地解决由时间依赖的非线性部分差异方程式所决定的、以时间依赖的非线性最佳控制问题。我们的目标是提供一个一般工具,处理许多冰河环境中若干非线性最佳系统的时间演变问题,必须对各种物理和几何特征进行系统分析。最佳控制是一种工具,可以用来填补所收集的数据和数学模型之间的空白,通常与非常耗时的活动有关:反向问题、统计等。标准离散技术可能导致无法忍受的真实应用模拟。我们的目标是展示如何降低排序模型能够解决这个问题。我们依靠一个空间-时间POD-Galerkin的缩减,以便在低维度缩小的空间快速地解决最佳控制问题,以几处参数为例。拟议算法的一般性得到基于环境科学的数字测试的验证:减少最佳控制问题,不仅在物理特征中,而且在几何物理特征中都受到调整。我们的目标是展示如何以更快的速度恢复模型的准确度。我们将显示如何以更高的速度恢复模型。