Optimal Control in Thermo-Electro-Elasticity: AddressingSignorini’s Problemwith Tresca’s Friction Law

This study focuses on investigating an optimal control problem that represents the unilateral frictional contact between a thermo-electro-elastic body and a foundation with electrical and thermal conductivity. The problem is considered in a static setting, and the material behavior is described using linear thermo-piezoelectric constitutive laws. Tresca’s friction law is used to model the contact, along with regularized conditions for electrical and thermal conductivity. The mathematical properties of the problem, including its variational formulation, existence, and uniqueness of weak solutions, are discussed. Additionally, an optimal control problem is formulated, and the existence of at least one solution is established. A regularized version of the unilateral contact and Tresca’s law is then incorporated into the model. The existence of a weak solution to the regularized control problem is proven, and its convergence to the solution of the original optimal control problem is verified. Finally, an optimality condition is presented for this type of problem.