Applied Mathematics & Optimization

, Volume 78, Issue 2, pp 379–401

# Boundary Optimal Control for a Frictional Contact Problem with Normal Compliance

• Andaluzia Matei
• Sorin Micu
Article

## Abstract

We consider the contact between an elastic body and a deformable foundation. Firstly, we introduce a mathematical model for this phenomenon by means of a normal compliance contact condition associated with a friction law. Then, we propose a variational formulation of the model in a form of a quasi-variational inequality governed by a non-differentiable functional and we briefly discuss its well-possedness. Nextly, we address an optimal control problem related to this model in order to led the displacement field as close as possible to a given target by acting with a localized boundary control. By using some mollifiers of the normal compliance functions, we introduce a regularized model which allows us to establish an optimality condition. Finally, by means of asymptotic analysis tools, we show that the solutions of the regularized optimal control problems converge to a solution of the initial optimal control problem.

## Keywords

Frictional contact Elastic body Deformable foundation Normal compliance Quasi-variational inequality Weak solution Boundary optimal control Regularization Optimality condition

## Mathematics Subject Classification

74M10 74M15 49J20 49K20

## Notes

### Acknowledgements

This work was supported by a Grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, Project number PN-II-ID-PCE-2011-3-0257 and by a LEA MATH-MODE Project CNRS-IMAR.

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