Our objective was to match, in an L2 sense, the temperature at the top of the solid over the interval 0.08 < x<1 to a given constant temperature. That given constant temperature is represented by the horizontal line T= 1.5 in the figure.
The upper curve in the figure, i.e., the one that deviates the most from the objective curve T= 1.5, is for the prescribed baseline control T= 1 at the fluid inflow.
The other two curves are the temperature at the top of the solid corresponding to the optimal controls. The curve that is closest to the objective T= 1.5 correponds to the choice of parameters in the functional that allows for the greatest amount of control.
Note that optimal solution for the temperature on the top of the solid is substantially closer to the objective temperature compared to the baseline temperature. This optimal state is determined as part of the solution of the coupled optimality system of partial differential equations. In this manner, optimal states are determined systematically without the need for intuition.
The control: the inflow boundary temperature