Abstract
In the present paper, we analyze and study the control of a static thermoelastic contact problem. We consider a model which describes a frictional contact problem between a thermoelastic body and a deformable heat conductor obstacle. We derive a variational formulation of the model which is in the form of a coupled system of the quasi-variational inequality of elliptic type for the displacement and the nonlinear variational equation for the temperature. Then, under a smallness assumption, we prove the existence of a unique weak solution to the problem. Moreover, we establish the dependence of the solution with respect to the data and prove a convergence result. Finally, we introduce an optimization problem related to the contact model for which we prove the existence of a minimizer and provide a convergence result.
Highlights
The study of contact problems involving thermo-elastic materials remains an active research area
We can see [2,3,4, 6,7,8] for general thermoelastic models and their analysis, [9, 14, 15, 16, 21, 22] for the mathematical treatment of optimal control for a system governed by variational equations and inequalities
We refer to [1, 11, 12, 13, 17, 19, 20] and more recently [5,18] for some comprehensive references on analysis optimal control problems arising from contact models
Summary
The study of contact problems involving thermo-elastic materials remains an active research area. Due to the intrinsic coupling between mechanical and thermal energy, these materials has attracted the attention of industry and engineering researchers. For this reason, a considerable effort has been made in its modelling and numerical simulations of contact problems, and the literature concerning this topic is rather extensive. After proving the unique weak solvability of the contact problem, as well as a convergence result of the solution with respect to the data, we consider an optimization problem related to our contact problem, for which we provide under some smallness conditions, the existence of a minimizer and a convergence result. We give two examples of optimization problems that illustrate our results
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