Optimization of Thermodynamic Processes in Heat-Conducting Rods Controlled by a Peltier Element

An optimization problem of heat transfer in two thin rods, which are separated by a Peltier element inducing a control heat flow between them, is under study. The considered model takes into account energy exchange with the environment, intrinsic thermal conductivity of the Peltier element and thermal insulation at the free ends of the rods. Given an initial temperature distribution and a fixed time horizon, the time history of control heat flux is sought for cooling or heating one of the rods to the ambient temperature. To optimize the control, an integral statement of constitutive relations, namely the Fourier law and heat exchange with air, is applied. In the numerical simulation of thermodynamic process, the finite element method is used and regularization of the control problem is performed by adding the constitutive integral to the cost functional.