In this work we deal with parametrized time dependent optimal control problems governed by partial differential equations. We aim at extending the standard saddle point framework of steady constraints to time dependent cases. We provide an analysis of the well-posedness of this formulation both for parametrized scalar parabolic constraint and Stokes governing equations and we propose reduced order methods as an effective strategy to solve them. Indeed, on one hand, parametrized time dependent optimal control is a very powerful mathematical model which is able to describe several physical phenomena, on the other, it requires a huge computational effort. Reduced order methods are a suitable approach to have rapid and accurate simulations. We rely on POD–Galerkin reduction over the physical and geometrical parameters of the optimality system in a space-time formulation. Our theoretical results and our methodology are tested on two examples: a boundary time dependent optimal control for a Graetz flow and a distributed optimal control governed by time dependent Stokes equations. With these two test cases the convenience of the reduced order modelling is further extended to the field of time dependent optimal control.

在这项工作中，我们处理由偏微分方程控制的参数化时间相关的最优控制问题。我们旨在将稳定约束的标准鞍点框架扩展到时间相关的情况。我们对参数化标量抛物线约束和Stokes控制方程都提供了该公式的适定性分析，并提出了降阶方法作为解决它们的有效策略。实际上，一方面，参数化的时间相关最优控制是一个非常强大的数学模型，它能够描述几种物理现象，另一方面，它需要大量的计算工作。降阶方法是进行快速而准确的仿真的合适方法。在时空公式中，我们依靠POD–Galerkin减少最优系统的物理和几何参数。我们的理论结果和方法论在两个示例上进行了测试：Graetz流的边界时间依赖的最优控制和受时间依赖的Stokes方程控制的分布式最优控制。通过这两个测试用例，降阶建模的便利性进一步扩展到了与时间有关的最佳控制领域。