In this paper we will consider distributed Linear-Quadratic Optimal Control Problems dealing with Advection-Diffusion PDEs for high values of the P\'eclet number. In this situation, computational instabilities occur, both for steady and unsteady cases. A Streamline Upwind Petrov-Galerkin technique is used in the optimality system to overcome these unpleasant effects. We will apply a finite element method discretization in an optimize-then-discretize approach. Concerning the parabolic case, a stabilized space-time framework will be considered and stabilization will also occur in both bilinear forms involving time derivatives. Then we will build Reduced Order Models on this discretization procedure and two possible settings can be analyzed: whether or not stabilization is needed in the online phase, too. In order to build the reduced bases for state, control, and adjoint variables we will consider a Proper Orthogonal Decomposition algorithm in a partitioned approach. It is the first time that Reduced Order Models are applied to stabilized parabolic problems in this setting. The discussion is supported by computational experiments, where relative errors between the FEM and ROM solutions are studied together with the respective computational times.

翻译：本文研究高Péclet数下基于对流-扩散偏微分方程的分布型线性二次最优控制问题。在此情况下，稳态与非稳态问题均会出现计算不稳定性。最优性系统中采用流线迎风Petrov-Galerkin技术以克服这些不良效应。我们采用“先优化后离散”的方法进行有限元法离散。针对抛物型情况，将考虑稳定的时空框架，并在涉及时间导数的双线性形式中同样引入稳定化处理。随后基于该离散过程构建降阶模型，并分析两种可能设置：在线阶段是否需要额外稳定化。为构建状态、控制及伴随变量的降阶基，我们采用一种分块式的本征正交分解算法。这是首次将降阶模型应用于此类稳定的抛物型问题。计算实验验证了相关讨论，同时对比了有限元解与降阶模型解之间的相对误差及计算时间。