Optimal Control of Steady-State Heat Transfer in Two Solids with Continuous Flow and Thermal Jump at the Interface

This work presents a distributed optimal control problem for steady-state heat conduction in a system made up of two solids in thermal contact, with heat flux continuity and a temperature jump at the interface. The control affects the system’s energy source. The existence and uniqueness of optimal control are established, and the corresponding optimality conditions are derived. Additionally, it is shown that optimal control can be considered a fixed point of a well-defined operator. Moreover, an iterative algorithm is introduced to approximate the solution to the optimal control problem, which converges regardless of the initial data. Finally, an explicit solution related to one-dimensional case in Cartesian coordinates is given.