The horizontal line T = 1 at the fluid inflow is a prescribed control that gives a baseline for comparison to optimal solutions. Note that control is only applied at the fluid inflow 0 < y< 0.75 and not at the left end of the solid 0.75 < y< 1.
The other two curves represent optimal controls for two different set of values for the wieghts in the penalized objective functional. The weights corresponding the curve that starts at lowest at the left end are such that a larger control is available for use in meeting the objective.
Note that the optimal control, i.e., the fluid inflow temperature, varies substantially from the baseline temperature T = 1. The optimal control is determined as part of the solution of the coupled optimality system of partial differential equations. In this manner, optimal control are determined systematically without the need for intuition.
The objective: the temperature at the top of the solid