Analysis and control of an electro-elastic contact problem

We consider a mathematical model that describes the frictional contact of an electro-elastic body with a semi-insulator foundation. The process is static; the contact is bilateral and associated to Tresca’s friction law. We list the assumptions on the data and derive a variational formulation of the model, in the form of a system that couples two inclusions in which the unknowns are the strain field and the electric field. Then we prove the unique solvability of the system, as well as the continuous dependence of its solution with respect to the data. We use these results in the study of an associated optimal control problem, for which we prove an existence result. The proofs are based on arguments of monotonicity, compactness, convex analysis, and lower semicontinuity.