Optimal Control for the Navier–Stokes Equation with Time Delay in the Convection: Analysis and Finite Element Approximations

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.