Optimal control of a frictional thermo-piezoelectric contact problem

Abstract

In this paper, we provide an optimal control, necessary optimality conditions and a convergence result of regularized problem for a mathematical model which describes frictional contact between a thermo-electro-elastic body and an electrically and thermally conductive foundation. The contact is described by Tresca’s friction law including electrical and thermal conductivity conditions. We derive a variational formulation of our model which is given as a coupled system for the displacement, electric potential, and the temperature fields. Then, an existence and uniqueness theorem is established. Our aim here is to present a detailed description to controlling the corresponding deformation and electrical potential in the body, when the foundation temperature is the control. To do this, we introduce an optimal control problem and under some regularity assumptions, we prove the existence of at least one solution. Moreover, we introduce the relate regularized problem and we show the dependence of this problem with respect to the data and we provide a convergence result. Finally, we provide a necessary optimality condition of the regularized optimal control problem.