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Convergence and Optimization Results for a History-Dependent Variational Problem

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Abstract

We consider a mixed variational problem in real Hilbert spaces, defined on the unbounded interval of time \([0,+\infty)\) and governed by a history-dependent operator. We state the unique solvability of the problem, which follows from a general existence and uniqueness result obtained in Sofonea and Matei (J. Glob. Optim. 61:591–614, 2015). Then, we state and prove a general convergence result. The proof is based on arguments of monotonicity, compactness, lower semicontinuity and Mosco convergence. Finally, we consider a general optimization problem for which we prove the existence of minimizers. The mathematical tools developed in this paper are useful in the analysis of a large class of nonlinear boundary value problems which, in a weak formulation, lead to history-dependent mixed variational problems. To provide an example, we illustrate our abstract results in the study of a frictional contact problem for viscoelastic materials with long memory.

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Acknowledgements

This project has received funding from 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. Laboratoire de Mathématiques et Physique, Université de Perpignan Via Domitia, 52 Avenue Paul Alduy, 66860, Perpignan, France

    Mircea Sofonea

  2. Department of Mathematics, University of Craiova, A.I. Cuza Street 13, 200585, Craiova, Romania

    Andaluzia Matei

Authors
  1. Mircea Sofonea
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  2. Andaluzia Matei
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Correspondence to Mircea Sofonea.

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Sofonea, M., Matei, A. Convergence and Optimization Results for a History-Dependent Variational Problem. Acta Appl Math 169, 157–182 (2020). https://doi.org/10.1007/s10440-019-00293-x

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