A distributed optimal control problem for the 2D incompressible Navier–Stokes equation with delay in the convection term is studied. The delay corresponds to the non-instantaneous effect of the motion of a fluid parcel on the mass transfer, and can be realized as a regularization or stabilization to the Navier–Stokes equation. The existence of optimal controls is established, and the corresponding first-order necessary optimality system is determined. A semi-implicit discontinuous Galerkin scheme with respect to time and conforming finite elements for space is considered. Error analysis for this numerical scheme is discussed and optimal convergence rates are proved. The fully discrete problem is solved by the Barzilai-Borwein gradient method. Numerical examples for the velocity-tracking and vorticity minimization problems based on the Taylor-Hood elements are presented.

研究了对流项中带时滞的二维不可压缩Navier-Stokes方程的分布式最优控制问题。延迟对应于流体包裹运动对传质的非瞬时影响，可以通过对Navier–Stokes方程进行正则化或稳定化来实现。建立最优控制的存在，并确定相应的一阶必要最优系统。考虑了关于时间和空间的有限元的半隐式不连续Galerkin方案。讨论了该数值方案的误差分析，并证明了最佳收敛速度。完全离散的问题通过Barzilai-Borwein梯度法解决。