| Title:
|
Optimal control of a frictionless contact problem with normal compliance (English) |
| Author:
|
Touzaline, Arezki |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
59 |
| Issue:
|
3 |
| Year:
|
2018 |
| Pages:
|
327-342 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We consider a mathematical model which describes a contact between an elastic body and a foundation. The contact is frictionless with normal compliance. The goal of this paper is to study an optimal control problem which consists of leading the stress tensor as close as possible to a given target, by acting with a control on the boundary of the body. We state an optimal control problem which admits at least one solution. Next, we establish an optimality condition corresponding to a regularization of the model. We also introduce the regularized control problem for which we study the convergence when the regularization parameter tends to zero. (English) |
| Keyword:
|
optimal control |
| Keyword:
|
variational inequality |
| Keyword:
|
linear elastic frictionless contact |
| Keyword:
|
regularized problem |
| MSC:
|
47J20 |
| MSC:
|
49J40 |
| MSC:
|
74M10 |
| idZBL:
|
Zbl 06940874 |
| idMR:
|
MR3861556 |
| DOI:
|
10.14712/1213-7243.2015.251 |
| . |
| Date available:
|
2018-09-10T12:12:29Z |
| Last updated:
|
2020-10-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147401 |
| . |
| Reference:
|
[1] Amassad A., Chenais D., Fabre C.: Optimal control of an elastic contact problem involving Tresca friction law.Nonlinear Anal. Ser. A: Theory Methods 48 (2002), no. 8, 1107–1135. MR 1880576 |
| Reference:
|
[2] Andersson L.-E.: A quasistatic frictional problem with normal compliance.Nonlinear Anal. 16 (1991), no. 4, 347–370. MR 1093846, 10.1016/0362-546X(91)90035-Y |
| Reference:
|
[3] Barbu V.: Optimal Control of Variational Inequalities.Research Notes in Mathematics, 100, Pitman (Advanced Publishing Program), Boston, 1984. Zbl 0696.49021, MR 0742624 |
| Reference:
|
[4] Bartosz K., Kalita P.: Optimal control for a class of dynamic viscoelastic contact problems with adhesion.Dynam. Systems Appl. 21 (2012), no. 2–3, 269–292. MR 2918380 |
| Reference:
|
[5] Bonnans J. F., Tiba D.: Pontryagin's principle in the control of semiliniar elliptic variational inequalities.Appl. Mathem. Optim. 23 (1991), no. 3, 299–312. MR 1095664, 10.1007/BF01442403 |
| Reference:
|
[6] Capatina A., Timofte C.: Boundary optimal control for quasistatic bilateral frictional contact problems.Nonlinear Anal. 94 (2014), 84–99. MR 3120675, 10.1016/j.na.2013.08.004 |
| Reference:
|
[7] Denkowski Z., Migórski S., Ochal A.: Optimal control for a class of mechanical thermoviscoelastic frictional contact problems.Control Cybernet. 36 (2007), no. 3, 611–632. MR 2376043 |
| Reference:
|
[8] Denkowski Z., Migórski S., Ochal A.: A class of optimal control problems for piezoelectric frictional contact models.Nonlinear Anal. Real World Appl. 12 (2011), no. 3, 1883–1895. MR 2781904 |
| Reference:
|
[9] Friedman A.: Optimal control for variational inequalities.SIAM J. Control Optim. 24 (1986), no. 3, 439–451. MR 0838049, 10.1137/0324025 |
| Reference:
|
[10] Kikuchi N., Oden J. T.: Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods.SIAM Studies in Applied Mathematics, 8, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1988. MR 0961258 |
| Reference:
|
[11] Kimmerle S.-J., Moritz R.: Optimal control of an elastic Tyre-Damper system with road contact.(P. Steinmann, G. Leugering, eds.) Special Issue: 85th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM), Erlangen 2014, 14 (2014), no. 1, 875–876. |
| Reference:
|
[12] Klarbring A., Mikelić A., Shillor M.: Frictional contact problems with normal compliance.Internat. J. Engrg. Sci. 26 (1988), no. 8, 811–832. MR 0958441 |
| Reference:
|
[13] Klarbring A., Mikelić A., Shillor M.: On frictional problems with normal compliance.Nonlinear Anal. 13 (1989), no. 8, 935–955. MR 1009079 |
| Reference:
|
[14] Laursen T. A.: Computational Contact and Impact Mechanics.Fundamentals of Modeling Interfacial Phenomena in Nonlinear Finite Element Analysis, Springer, Berlin, 2002. MR 1902698 |
| Reference:
|
[15] Lions J.-L.: Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles.Gauthier-Villars, Paris, 1968 (French). MR 0244606 |
| Reference:
|
[16] Martins J. A. C., Oden J. T.: Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws.Nonlinear Anal. 11 (1987), no. 3, 407–428. MR 0881727, 10.1016/0362-546X(87)90055-1 |
| Reference:
|
[17] Matei A., Micu S.: Boundary optimal control for nonlinear antiplane problems.Nonlinear Anal. 74 (2011), no. 5, 1641–1652. MR 2764365, 10.1016/j.na.2010.10.034 |
| Reference:
|
[18] Matei A., Micu S.: Boundary optimal control for a frictional problem with normal compliance.Appl. Math. Optim. (2017), 23 pages. |
| Reference:
|
[19] Mignot F.: Contrôle dans les inéquations variationnelles elliptiques.J. Functional Analysis 22 (1976), no. 2, 130–185 (French). MR 0423155, 10.1016/0022-1236(76)90017-3 |
| Reference:
|
[20] Mignot R., Puel J.-P.: Optimal control in some variational inequalities.SIAM J. Control Optim. 22 (1984), no. 3, 466–476. MR 0739836, 10.1137/0322028 |
| Reference:
|
[21] Neittaanmaki P., Sprekels J., Tiba D.: Optimization of Elliptic Systems: Theory and Applications.Springer Monographs in Mathematics, Springer, New York, 2006. MR 2183776 |
| Reference:
|
[22] Oden J. T., Martins J. A. C.: Models and computational methods for dynamic friction phenomena.Comput. Methods Appl. Mech. Engrg. 52 (1985), no. 1–3, 527–634. MR 0822757, 10.1016/0045-7825(85)90009-X |
| Reference:
|
[23] Popov V. L.: Contact Mechanics and Friction.Physical Principles and Applications, Springer, Heidelberg, 2010. |
| Reference:
|
[24] Rochdi M., Shillor M., Sofonea M.: Quasistatic viscoelastic contact with normal compliance and friction.J. Elasticity 51 (1998), no. 2, 105–126. MR 1664496, 10.1023/A:1007413119583 |
| Reference:
|
[25] Shillor M., Sofonea M., Telega J. J.: Models and Variational Analysis of Quasistatic Contact.Variational Methods, Lecture Notes in Physics, 655, Springer, Berlin, 2004. 10.1007/b99799 |
| Reference:
|
[26] Sofonea M., Matei A.: Variational Inequalities with Applications.A Study of Antiplane Frictional Contact Problems, Advances in Mechanics and Mathematics, 18, Springer, 2009. MR 2488869 |
| Reference:
|
[27] Sofonea M., Matei A.: Mathematical Models in Contact Mechanics.London Mathematical Society, Lecture Note Series, 398, Cambridge University Press, 2012. |
| Reference:
|
[28] Touzaline A.: Optimal control of a frictional contact problem.Acta Math. Appl. Sin. Engl. Ser. 31 (2015), no. 4, 991–1000. MR 3418276, 10.1007/s10255-015-0519-8 |
| . |
