Thermal control of the Souza-Auricchio model for shape memory alloys

We address the thermal control of the quasi-static evolution of a polycrystalline shape memory alloy specimen. The thermomechanical evolution of the body is described by means of the phenomenological Souza-Auricchio model. By assuming to be able to control the temperature of the body in time we determine the corresponding quasi-static evolution in the energetic sense. By recovering in this context a result by Rindler we prove the existence of optimal controls for a suitably large class of cost functionals and comment on their possible approximation.