The top figure is a plot of the target or reference velocity field at t= 0. We see that the reference flow starts from rest. The bottom figure is the intial condition (at t= 0) imposed on the optimal velocity field, i.e., candidate minimizers are required to satify this initial velocity distribution. The body force is then used to drive our velocity field, at subsequent times, towards the target velocity.
It is clear from the two figures that the required initial condition for the optimal solution is nowhere near that of the target velocity, i.e., we are not starting close to the target velocity.
The target and optimal velocity fields at t = 0.1
The target and optimal velocity fields at t = 0.3
The target and optimal velocity fields at t = 0.4
The target and optimal velocity fields at t = 0.5
The norm of the difference between the target and optimal velocities vs. time
The norm of the control vs. time