Optimal control of a contact problem with slip dependent friction

Abderrezak Kasri; Arezki Touzaline

doi:10.2478/awutm-2022-0003

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Optimal control of a contact problem with slip dependent friction

Annals of West University of Timisoara - Mathematics and Computer Science

Volume 58 (2022): Issue 1 (June 2022)

By: Abderrezak Kasri and Arezki Touzaline

Open Access

|Jun 2022

Abstract

The aim of this paper is to study a frictional contact problem between an elastic body and a foundation. We focus on the optimal control of the model which consists of acting with a control on a portion of the boundary of the body and leading the stress tensor as close as possible to a given target. We state an optimal control problem for which we establish an existence theorem of the solution. We then introduce a sequence of regularized problems depending on a positive parameter α and we study the convergence of the sequence of the solutions when the parameter α tends to zero. Finally, an optimality condition is established for this problem.

References

[1] A. Amassad, D. Chenais, C. Fabre, Optimal control of an elastic contact problem involving Tresca friction law, Nonlinear Analysis 48 (2002), 1107–1135.10.1016/S0362-546X(00)00241-8
Search in Google Scholar Back to article
[2] V. Barbu, Optimal Control of Variational Inequalities, Pitman Advanced Publishing, Boston, 1984.
Search in Google Scholar Back to article
[3] A. Bermudez, C. Saguez, Optimal control of a Signorini problem, SIAM. J. Control Optim. 25 (3) (1987), 576–582.10.1137/0325032
Search in Google Scholar Back to article
[4] A. Capatina, C. Timofte, An optimal control problem governed by implicit evolution quasi-variational inequalities, Annals of the University of Bucharest (Mathematical series) 4 (LXII) (2013), 157–166.
Search in Google Scholar Back to article
[5] G. Duvaut, J. L. Lions, Les inéquations en mécanique et en physique, Dunod, Paris, 1972.
Search in Google Scholar Back to article
[6] A. Friedman, Optimal Control for Variational Inequalities, SIAM, Journal on Control and Optimization 24 (3) (1986), 439–451.10.1137/0324025
Search in Google Scholar Back to article
[7] I. R. Ionescu, Q.-L. Nguyen, S. Wolf, Slip-dependent friction in dynamic elasticity, Nonlinear Analysis 53 (2003), 375–390.10.1016/S0362-546X(02)00302-4
Search in Google Scholar Back to article
[8] A. Kasri, A. Touzaline, A quasistatic frictional contact problem for viscoelastic materials with long memory, Georgian Math. J. 27 (2020), 249–264.10.1515/gmj-2018-0002
Search in Google Scholar Back to article
[9] J. L. Lions, Contrôle optimal des systèmes gouvernés par des équations aux dérivées partielles, Dunod, Paris, 1968.
Search in Google Scholar Back to article
[10] J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod et Gauthier-Villars, 1969.
Search in Google Scholar Back to article
[11] J. Lovišek, Optimal control of a variational inequality with possibly, nonsymmetric linear operator. Application to the obstacle problems in mathematical physics, Acta Math. Univ. Comenianae LXIII (1) (1994), 1–23.
Search in Google Scholar Back to article
[12] A. Matei, S. Micu, Boundary optimal control for nonlinear antiplane problems, Nonlinear Analysis: Theory Methods and Applications 74 (2011), 1641–1652.10.1016/j.na.2010.10.034
Search in Google Scholar Back to article
[13] A. Matei, S. Micu, Boundary optimal control for a frictional contact problem with normal compliance, Applied Mathematics and Optimization 78 (2018), 379–401.10.1007/s00245-017-9410-8
Search in Google Scholar Back to article
[14] R. Mignot, Contrôle dans les inéquations variationnelles elliptiques, J. Fun. Anal. 22 (1976), 130–185.10.1016/0022-1236(76)90017-3
Search in Google Scholar Back to article
[15] R. Mignot, J.-P. Puel, Optimal control in some variational inequalities, SIAM, J. Control Optim. 22 (1984), 466–476.10.1137/0322028
Search in Google Scholar Back to article
[16] J. Neĉas, I. Hlavaĉek, Mathematical Theory of Elastic and Elastoplastic Bodies: An Introduction, Elsevier, Amsterdam, 1981.
Search in Google Scholar Back to article
[17] P. Neittaanmaki, J. Sprekels, D. Tiba, Optimization of Elliptic Systems: Theory and Applications, Springer Monographs in Mathematics, Springer, New York, 2006.
Search in Google Scholar Back to article
[18] M. Shillor, M. Sofonea, J. J. Telega, Models and Analysis of Quasistatic Contact: Variational Methods, Lecture Notes in Physics 655, Springer, Berlin, 2004.10.1007/b99799
Search in Google Scholar Back to article
[19] A. Touzaline, Optimal control of a frictional contact problem, Acta Mathematicae Applicatae Sinica, English Series 31 (4) (2015), 1–10, https://doi.org/10.1007/s10255-015-0519-8.
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/awutm-2022-0003 | Journal eISSN: 1841-3307 | Journal ISSN: 1841-3293

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Language: English

Page range: 18 - 38

Published on: Jun 20, 2022

Published by: West University of Timisoara

In partnership with: Paradigm Publishing Services

Publication frequency: Volume open

Keywords:

optimal control,

variational inequalities,

elastic materials,

slip-dependent friction,

regularized problem

Related subjects:

Mathematics,

General mathematics

© 2022 Abderrezak Kasri, Arezki Touzaline, published by West University of Timisoara
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Volume 58 (2022): Issue 1 (June 2022)