A History-Dependent Inclusion with Applications in Contact Mechanics

We consider an abstract inclusion in a real Hilbert space, governed by an almost history-dependent operator. We prove the unique solvability of the inclusion as well as the continuous dependence of the solution with respect to the data. Then, we use these results to study an associated control problem for which we prove the existence of optimal control as well as a convergence result. The proofs are based on arguments of monotonicity, compactness, convex analysis and lower semicontinuity. We also consider a mathematical model which describes the equilibrium of a viscoelastic body in bilateral contact with a foundation. We derive a variational formulation of the model that is in a form of a history-dependent inclusion for the strain field. Then we apply our abstract results in the analysis and control of this contact model.