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（聚合方法与极大化器方法）的特性。两种方法均通过扩充降阶基空间来确保稳定性。对于聚合方法，状态变量与伴随变量的降阶基逼近空间被扩充至彼此相同。对于极大化器方法，状态-控制乘积空间的降阶基逼近空间通过求解极大化器方程获得的解进行扩充。我们分别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，为达到相同近似精度，极大化器方法所需的降阶基向量数量远多于聚合方法（可能达到两倍）。在某些情形下，极大化器方法甚至无法生成收敛解。本文通过对比两种稳定化方法的精度、效率及计算成本，证实聚合稳定化方法是控制问题中更具优越性的稳定化策略。