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Optimization problems for elastic contact models with unilateral constraints

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Abstract

The aim of this paper is to provide some results in the study of an abstract optimization problem in reflexive Banach spaces and to illustrate their use in the analysis and control of static contact problems with elastic materials. We start with a simple model problem which describes the equilibrium of an elastic body in unilateral contact with a foundation. We derive a variational formulation of the model which is in the form of minimization problem for the stress field. Then we introduce the abstract optimization problem for which we prove existence, uniqueness and convergence results. The proofs are based on arguments of lower semicontinuity, monotonicity, convexity, compactness and Mosco convergence. Finally, we use these abstract results to deduce both the unique solvability of the contact model and the existence and the convergence of the optimal pairs for an associated optimal control problem.

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Acknowledgements

This research was supported by the National Natural Science Foundation of China (11771067) and the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie Grant Agreement No 823731 CONMECH.

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, Sichuan, People’s Republic of China

    Mircea Sofonea & Yi-bin Xiao

  2. Laboratoire de Mathématiques et Physique, University of Perpignan Via Domitia, 52 Avenue Paul Alduy, 66860, Perpignan, France

    Mircea Sofonea & Maxime Couderc

Authors
  1. Mircea Sofonea
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  2. Yi-bin Xiao
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  3. Maxime Couderc
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Correspondence to Yi-bin Xiao.

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Sofonea, M., Xiao, Yb. & Couderc, M. Optimization problems for elastic contact models with unilateral constraints. Z. Angew. Math. Phys. 70, 1 (2019). https://doi.org/10.1007/s00033-018-1046-2

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  • DOI: https://doi.org/10.1007/s00033-018-1046-2

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