Optimal Control for a Dynamic Thermo-Electro-Viscoelastic Contact Problem with Frictional Heating

Abstract

This paper focuses on optimal control of a coupled system modeling a dynamic frictional thermo-electroviscoelastic contact problem between a piezoelectric body and an electrically and thermally conductive foundation. The material’s behavior is described by a linear thermo-viscoelectroelastic constitutive law and the contact is modeled with a compliance normal condition coupled with Coulomb’s friction law, an electric condition in the Tresca’s form and thermal condition taking into account both heat transfer and frictional heating process. The weak formulation of the problem consists of a system of two variational inequalities and nonlinear variational equations. We provide existence and uniqueness results of a weak solution to the model and, under some additional assumptions, the continuous dependence of a solution on the problem’s data. Finally, for a class of optimal control problems and inverse problems, we prove the existence of optimal solutions.