Optimal Control in Thermo-Electro-Elasticity: Addressing
Signorini’s Problem
with Tresca’s Friction Law
摘要
This study focuses on investigating an optimal control problem that represents theunilateral frictional contact between a thermo-electro-elastic body and a foundation with electricaland thermal conductivity. The problem is considered in a static setting, and the material behavioris described using linear thermo-piezoelectric constitutive laws. Tresca’s friction law is used tomodel the contact, along with regularized conditions for electrical and thermal conductivity. Themathematical properties of the problem, including its variational formulation, existence, anduniqueness of weak solutions, are discussed. Additionally, an optimal control problem isformulated, and the existence of at least one solution is established. A regularized version of theunilateral contact and Tresca’s law is then incorporated into the model. The existence of a weaksolution to the regularized control problem is proven, and its convergence to the solution of theoriginal optimal control problem is verified. Finally, an optimality condition is presented for thistype of problem.