In this paper, we consider a state constrained optimal control problem governed by the transient Stokes equations. The state constraint is given by an L2 functional in space, which is required to fulfill a pointwise bound in time. The discretization scheme for the Stokes equations consists of inf-sup stable finite elements in space and a discontinuous Galerkin method in time, for which we have recently established best approximation type error estimates. Using these error estimates, for the discrete control problem we establish error estimates and as a by-product we show an improved regularity for the optimal control. We complement our theoretical analysis with numerical results.

翻译：本文研究受瞬态Stokes方程控制且状态约束在时间上逐点成立的最优控制问题。状态约束由空间上的L2泛函给出，并要求在时间维度上满足逐点有界性。Stokes方程的离散化方案采用空间上满足inf-sup稳定条件的有限元及时向上的间断Galerkin方法，我们近期已为此类方法建立了最佳逼近型误差估计。基于这些误差估计结果，我们对离散控制问题建立了误差估计，并作为推论得到了最优控制改进的正则性结果。理论分析通过数值算例加以验证。