We explore the features of two methods of stabilization, aggregation and supremizer methods, for reduced-order modeling of parametrized optimal control problems. In both methods, the reduced basis spaces are augmented to guarantee stability. For the aggregation method, the reduced basis approximation spaces for the state and adjoint variables are augmented in such a way that the spaces are identical. For the supremizer method, the reduced basis approximation space for the state-control product space is augmented with the solutions of a supremizer equation. We implement both of these methods for solving several parametrized control problems and assess their performance. Results indicate that the number of reduced basis vectors needed to approximate the solution space to some tolerance with the supremizer method is much larger, possibly double, that for aggregation. There are also some cases where the supremizer method fails to produce a converged solution. We present results to compare the accuracy, efficiency, and computational costs associated with both methods of stabilization which suggest that stabilization by aggregation is a superior stabilization method for control problems.

翻译：本文探讨了参数化最优控制问题降阶建模中两种稳定化方法——聚合方法与戴帽方法的特性。两种方法均通过扩充降阶基空间来保证稳定性。在聚合方法中，状态变量与伴随变量的降阶基逼近空间被扩充至完全相同的形式；在戴帽方法中，状态-控制乘积空间的降阶基逼近空间通过求解戴帽方程的解进行扩充。我们采用这两种方法求解多个参数化控制问题并评估其性能。结果表明，戴帽法在逼近求解空间至给定容差时所需降阶基向量数量远大于（可能达到两倍）聚合方法，且存在戴帽法无法收敛解的案例。我们通过对比两种稳定化方法的精度、效率及计算成本，表明聚合稳定化是控制问题中更优的稳定化方法。