# American Institute of Mathematical Sciences

• Previous Article
Convergence rate of strong approximations of compound random maps, application to SPDEs
• DCDS-B Home
• This Issue
• Next Article
Theoretical analysis on a diffusive SIR epidemic model with nonlinear incidence in a heterogeneous environment
December  2018, 23(10): 4477-4498. doi: 10.3934/dcdsb.2018172

## Convergence of solutions to inverse problems for a class of variational-hemivariational inequalities

 1 College of Applied Mathematics, Chengdu University of Information Technology, Chengdu, 610225, Sichuan Province, China 2 Jagiellonian University in Krakow, Faculty of Mathematics and Computer Science, Chair of Optimization and Control, ul. Lojasiewicza 6, 30-348 Krakow, Poland

* Corresponding author: biao.zeng@outlook.com

Received  May 2017 Revised  January 2018 Published  June 2018

Fund Project: The research is supported by the National Science Center of Poland under Maestro Project No. UMO-2012/06/A/ST1/00262, Special Funds of Guangxi Distinguished Experts Construction Engineering, Guangxi, China, and the International Project cofinanced by the Ministry of Science and Higher Education of Republic of Poland under Grant No. 3792/GGPJ/H2020/2017/0

The paper investigates an inverse problem for a stationary variational-hemivariational inequality. The solution of the variational-hemivariational inequality is approximated by its penalized version. We prove existence of solutions to inverse problems for both the initial inequality problem and the penalized problem. We show that optimal solutions to the inverse problem for the penalized problem converge, up to a subsequence, when the penalty parameter tends to zero, to an optimal solution of the inverse problem for the initial variational-hemivariational inequality. The results are illustrated by a mathematical model of a nonsmooth contact problem from elasticity.

Citation: Stanisław Migórski, Biao Zeng. Convergence of solutions to inverse problems for a class of variational-hemivariational inequalities. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4477-4498. doi: 10.3934/dcdsb.2018172
##### References:
 [1] B. Barabasz, S. Migórski, R. Schaefer and M. Paszynski, Multi deme, twin adaptive strategy $hp$-HGS, Inverse Problems in Science and Engineering, 19 (2011), 3-16. doi: 10.1080/17415977.2010.531477. Google Scholar [2] B. Barabasz, E. Gajda-Zagorska, S. Migórski, M. Paszynski, R. Schaefer and M. Smolka, A hybrid algorithm for solving inverse problems in elasticity, International Journal of Applied Mathematics and Computer Science, 24 (2014), 865-886. doi: 10.2478/amcs-2014-0064. Google Scholar [3] F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983. Google Scholar [4] Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory, Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003. doi: 10.1007/978-1-4419-9158-4. Google Scholar [5] Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Applications, Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003. Google Scholar [6] M. S. Gockenbach and A. A. Khan, An abstract framework for elliptic inverse problems. Ⅰ. an output least-squares approach, Math. Mech. Solids, 12 (2007), 259-276. doi: 10.1177/1081286505055758. Google Scholar [7] J. Gwinner, B. Jadamba, A. A. Khan and M. Sama, Identification in variational and quasi-variational inequalities, J. Convex Analysis, 25 (2018), 1-25. Google Scholar [8] W. Han, S. Migórski and M. Sofonea, A class of variational-hemivariational inequalities with applications to frictional contact problems, SIAM Journal of Mathematical Analysis, 46 (2014), 3891-3912. doi: 10.1137/140963248. Google Scholar [9] A. Hasanov, Inverse coefficient problems for potential operators, Inverse Problems, 13 (1997), 1265-1278. doi: 10.1088/0266-5611/13/5/011. Google Scholar [10] M. Hintermüller, Inverse coefficient problems for variational inequalities: Optimality conditions and numerical realization, M2AN Math. Model. Numer. Anal., 35 (2001), 129-152. doi: 10.1051/m2an:2001109. Google Scholar [11] B. Jadamba, A. A. Khan and M. Sama, Inverse problems of parameter identification in partial differential equations, in Mathematics in Science and Technology, World Sci. Publ., Hackensack, NJ, 2011,228-258. doi: 10.1142/9789814338820_0009. Google Scholar [12] V. K. Le, A range and existence theorem for pseudomonotone perturbations of maximal monotone operators, Proceedings of the American Mathematical Society, 139 (2011), 1645-1658. doi: 10.1090/S0002-9939-2010-10594-4. Google Scholar [13] Z. H. Liu and B. Zeng, Optimal control of generalized quasi-variational hemivariational inequalities and its applications, Appl. Math. Optim., 72 (2015), 305-323. doi: 10.1007/s00245-014-9281-1. Google Scholar [14] S. Manservisi and M. Gunzburger, A variational inequality formulation of an inverse elasticity problem, Applied Numerical Mathematics, 34 (2000), 99-126. doi: 10.1016/S0168-9274(99)00042-2. Google Scholar [15] S. Migórski, Identification of nonlinear heat transfer laws in problems modeled by hemivariational inequalities, in Inverse Problems in Engineering Mechanics, (eds. M. Tanaka and G. S. Dulikravich), Elsevier, 1998, 27-36. doi: 10.1016/B978-008043319-6/50007-8. Google Scholar [16] S. Migórski, Sensitivity analysis of inverse problems with applications to nonlinear systems, Dynamic Systems and Applications, 8 (1999), 73-88. Google Scholar [17] S. Migórski, Identification coefficient problems for elliptic hemivariational inequalities and applications, in Inverse Problems in Engineering Mechanics II (eds. M. Tanaka and G. S. Dulikravich), Elsevier, 2000.Google Scholar [18] S. Migórski, Homogenization technique in inverse problems for boundary hemivariational inequalities, Inverse Problems in Engineering, 11 (2003), 229-242. Google Scholar [19] S. Migórski, Identification of operators in systems governed by second order evolution inclusions with applications to hemivariational inequalities, International Journal of Innovative Computing, Information and Control, 8 (2012), 3845-3862. Google Scholar [20] S. Migórski and A. Ochal, Inverse coefficient problem for elliptic hemivariational inequality, in Nonsmooth/Nonconvex Mechanics, Modeling, Analysis and Numerical Methods (eds. D. Y. Gao et al.), Kluwer Academic Publishers, 50 (2001), 247-261. doi: 10.1007/978-1-4613-0275-9_11. Google Scholar [21] S. Migórski and A. Ochal, An inverse coefficient problem for a parabolic hemivariational inequality, Applicable Analysis, 89 (2010), 243-256. doi: 10.1080/00036810902889559. Google Scholar [22] S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics, 26 Springer, New York, 2013. doi: 10.1007/978-1-4614-4232-5. Google Scholar [23] S. Migórski, A. Ochal and M. Sofonea, A class of variational-hemivariational inequalities in reflexive Banach spaces, J. Elasticity, 127 (2017), 151-178. doi: 10.1007/s10659-016-9600-7. Google Scholar [24] S. Migórski and B. Zeng, Variational-hemivariational inverse problems for unilateral frictional contact, Applicable Analysis, (2018). doi: 10.1080/00036811.2018.1491037. Google Scholar [25] D. Motreanu and M. Sofonea, Quasivariational inequalities and applications in frictional contact problems with normal compliance, Adv. Math. Sci. Appl., 10 (2000), 103-118. Google Scholar [26] Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Marcel Dekker, Inc., New York, Basel, Hong Kong, 1995. Google Scholar [27] P. D. Panagiotopoulos, Hemivariational Inequalities, Applications in Mechanics and Engineering, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-51677-1. Google Scholar [28] M. Sofonea, W. Han and S. Migórski, Numerical analysis of history-dependent variational inequalities with applications to contact problems, European Journal of Applied Mathematics, 26 (2015), 427-452. doi: 10.1017/S095679251500011X. Google Scholar [29] M. Sofonea and A. Matei, Mathematical Models in Contact Mechanics, London Mathematical Society Lecture Note Series, 398 Cambridge University Press, 2012. doi: 10.1017/CBO9781139104166. Google Scholar [30] M. Sofonea, S. Migórski, Variational-Hemivariational Inequalities with Applications, Chapman & Hall/CRC, Monographs and Research Notes in Mathematics, Boca Raton, 2017.Google Scholar [31] M. Sofonea and F. Patrulescu, Penalization of history-dependent variational inequalities, European Journal of Applied Mathematics, 25 (2014), 155-176. doi: 10.1017/S0956792513000363. Google Scholar [32] E. Zeidler, Nonlinear Functional Analysis and Applications II A/B, Springer, New York, 1990. doi: 10.1007/978-1-4612-0985-0. Google Scholar

show all references

##### References:
 [1] B. Barabasz, S. Migórski, R. Schaefer and M. Paszynski, Multi deme, twin adaptive strategy $hp$-HGS, Inverse Problems in Science and Engineering, 19 (2011), 3-16. doi: 10.1080/17415977.2010.531477. Google Scholar [2] B. Barabasz, E. Gajda-Zagorska, S. Migórski, M. Paszynski, R. Schaefer and M. Smolka, A hybrid algorithm for solving inverse problems in elasticity, International Journal of Applied Mathematics and Computer Science, 24 (2014), 865-886. doi: 10.2478/amcs-2014-0064. Google Scholar [3] F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983. Google Scholar [4] Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory, Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003. doi: 10.1007/978-1-4419-9158-4. Google Scholar [5] Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Applications, Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003. Google Scholar [6] M. S. Gockenbach and A. A. Khan, An abstract framework for elliptic inverse problems. Ⅰ. an output least-squares approach, Math. Mech. Solids, 12 (2007), 259-276. doi: 10.1177/1081286505055758. Google Scholar [7] J. Gwinner, B. Jadamba, A. A. Khan and M. Sama, Identification in variational and quasi-variational inequalities, J. Convex Analysis, 25 (2018), 1-25. Google Scholar [8] W. Han, S. Migórski and M. Sofonea, A class of variational-hemivariational inequalities with applications to frictional contact problems, SIAM Journal of Mathematical Analysis, 46 (2014), 3891-3912. doi: 10.1137/140963248. Google Scholar [9] A. Hasanov, Inverse coefficient problems for potential operators, Inverse Problems, 13 (1997), 1265-1278. doi: 10.1088/0266-5611/13/5/011. Google Scholar [10] M. Hintermüller, Inverse coefficient problems for variational inequalities: Optimality conditions and numerical realization, M2AN Math. Model. Numer. Anal., 35 (2001), 129-152. doi: 10.1051/m2an:2001109. Google Scholar [11] B. Jadamba, A. A. Khan and M. Sama, Inverse problems of parameter identification in partial differential equations, in Mathematics in Science and Technology, World Sci. Publ., Hackensack, NJ, 2011,228-258. doi: 10.1142/9789814338820_0009. Google Scholar [12] V. K. Le, A range and existence theorem for pseudomonotone perturbations of maximal monotone operators, Proceedings of the American Mathematical Society, 139 (2011), 1645-1658. doi: 10.1090/S0002-9939-2010-10594-4. Google Scholar [13] Z. H. Liu and B. Zeng, Optimal control of generalized quasi-variational hemivariational inequalities and its applications, Appl. Math. Optim., 72 (2015), 305-323. doi: 10.1007/s00245-014-9281-1. Google Scholar [14] S. Manservisi and M. Gunzburger, A variational inequality formulation of an inverse elasticity problem, Applied Numerical Mathematics, 34 (2000), 99-126. doi: 10.1016/S0168-9274(99)00042-2. Google Scholar [15] S. Migórski, Identification of nonlinear heat transfer laws in problems modeled by hemivariational inequalities, in Inverse Problems in Engineering Mechanics, (eds. M. Tanaka and G. S. Dulikravich), Elsevier, 1998, 27-36. doi: 10.1016/B978-008043319-6/50007-8. Google Scholar [16] S. Migórski, Sensitivity analysis of inverse problems with applications to nonlinear systems, Dynamic Systems and Applications, 8 (1999), 73-88. Google Scholar [17] S. Migórski, Identification coefficient problems for elliptic hemivariational inequalities and applications, in Inverse Problems in Engineering Mechanics II (eds. M. Tanaka and G. S. Dulikravich), Elsevier, 2000.Google Scholar [18] S. Migórski, Homogenization technique in inverse problems for boundary hemivariational inequalities, Inverse Problems in Engineering, 11 (2003), 229-242. Google Scholar [19] S. Migórski, Identification of operators in systems governed by second order evolution inclusions with applications to hemivariational inequalities, International Journal of Innovative Computing, Information and Control, 8 (2012), 3845-3862. Google Scholar [20] S. Migórski and A. Ochal, Inverse coefficient problem for elliptic hemivariational inequality, in Nonsmooth/Nonconvex Mechanics, Modeling, Analysis and Numerical Methods (eds. D. Y. Gao et al.), Kluwer Academic Publishers, 50 (2001), 247-261. doi: 10.1007/978-1-4613-0275-9_11. Google Scholar [21] S. Migórski and A. Ochal, An inverse coefficient problem for a parabolic hemivariational inequality, Applicable Analysis, 89 (2010), 243-256. doi: 10.1080/00036810902889559. Google Scholar [22] S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics, 26 Springer, New York, 2013. doi: 10.1007/978-1-4614-4232-5. Google Scholar [23] S. Migórski, A. Ochal and M. Sofonea, A class of variational-hemivariational inequalities in reflexive Banach spaces, J. Elasticity, 127 (2017), 151-178. doi: 10.1007/s10659-016-9600-7. Google Scholar [24] S. Migórski and B. Zeng, Variational-hemivariational inverse problems for unilateral frictional contact, Applicable Analysis, (2018). doi: 10.1080/00036811.2018.1491037. Google Scholar [25] D. Motreanu and M. Sofonea, Quasivariational inequalities and applications in frictional contact problems with normal compliance, Adv. Math. Sci. Appl., 10 (2000), 103-118. Google Scholar [26] Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Marcel Dekker, Inc., New York, Basel, Hong Kong, 1995. Google Scholar [27] P. D. Panagiotopoulos, Hemivariational Inequalities, Applications in Mechanics and Engineering, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-51677-1. Google Scholar [28] M. Sofonea, W. Han and S. Migórski, Numerical analysis of history-dependent variational inequalities with applications to contact problems, European Journal of Applied Mathematics, 26 (2015), 427-452. doi: 10.1017/S095679251500011X. Google Scholar [29] M. Sofonea and A. Matei, Mathematical Models in Contact Mechanics, London Mathematical Society Lecture Note Series, 398 Cambridge University Press, 2012. doi: 10.1017/CBO9781139104166. Google Scholar [30] M. Sofonea, S. Migórski, Variational-Hemivariational Inequalities with Applications, Chapman & Hall/CRC, Monographs and Research Notes in Mathematics, Boca Raton, 2017.Google Scholar [31] M. Sofonea and F. Patrulescu, Penalization of history-dependent variational inequalities, European Journal of Applied Mathematics, 25 (2014), 155-176. doi: 10.1017/S0956792513000363. Google Scholar [32] E. Zeidler, Nonlinear Functional Analysis and Applications II A/B, Springer, New York, 1990. doi: 10.1007/978-1-4612-0985-0. Google Scholar
Outline of the paper
 [1] Stanisław Migórski. A note on optimal control problem for a hemivariational inequality modeling fluid flow. Conference Publications, 2013, 2013 (special) : 545-554. doi: 10.3934/proc.2013.2013.545 [2] Leszek Gasiński. Optimal control problem of Bolza-type for evolution hemivariational inequality. Conference Publications, 2003, 2003 (Special) : 320-326. doi: 10.3934/proc.2003.2003.320 [3] Junfeng Yang. Dynamic power price problem: An inverse variational inequality approach. Journal of Industrial & Management Optimization, 2008, 4 (4) : 673-684. doi: 10.3934/jimo.2008.4.673 [4] Changjun Yu, Kok Lay Teo, Liansheng Zhang, Yanqin Bai. On a refinement of the convergence analysis for the new exact penalty function method for continuous inequality constrained optimization problem. Journal of Industrial & Management Optimization, 2012, 8 (2) : 485-491. doi: 10.3934/jimo.2012.8.485 [5] Ming Chen, Chongchao Huang. A power penalty method for a class of linearly constrained variational inequality. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1381-1396. doi: 10.3934/jimo.2018012 [6] Stanislaw Migórski. Hemivariational inequality for a frictional contact problem in elasto-piezoelectricity. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1339-1356. doi: 10.3934/dcdsb.2006.6.1339 [7] Zhenhai Liu, Stanislaw Migórski. Noncoercive damping in dynamic hemivariational inequality with application to problem of piezoelectricity. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 129-143. doi: 10.3934/dcdsb.2008.9.129 [8] G. M. de Araújo, S. B. de Menezes. On a variational inequality for the Navier-Stokes operator with variable viscosity. Communications on Pure & Applied Analysis, 2006, 5 (3) : 583-596. doi: 10.3934/cpaa.2006.5.583 [9] X. X. Huang, D. Li, Xiaoqi Yang. Convergence of optimal values of quadratic penalty problems for mathematical programs with complementarity constraints. Journal of Industrial & Management Optimization, 2006, 2 (3) : 287-296. doi: 10.3934/jimo.2006.2.287 [10] Hang-Chin Lai, Jin-Chirng Lee, Shuh-Jye Chern. A variational problem and optimal control. Journal of Industrial & Management Optimization, 2011, 7 (4) : 967-975. doi: 10.3934/jimo.2011.7.967 [11] S. J. Li, Z. M. Fang. On the stability of a dual weak vector variational inequality problem. Journal of Industrial & Management Optimization, 2008, 4 (1) : 155-165. doi: 10.3934/jimo.2008.4.155 [12] Changjie Fang, Weimin Han. Well-posedness and optimal control of a hemivariational inequality for nonstationary Stokes fluid flow. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5369-5386. doi: 10.3934/dcds.2016036 [13] Zhili Ge, Gang Qian, Deren Han. Global convergence of an inexact operator splitting method for monotone variational inequalities. Journal of Industrial & Management Optimization, 2011, 7 (4) : 1013-1026. doi: 10.3934/jimo.2011.7.1013 [14] Vladimir E. Fedorov, Natalia D. Ivanova. Identification problem for a degenerate evolution equation with overdetermination on the solution semigroup kernel. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 687-696. doi: 10.3934/dcdss.2016022 [15] Martin Brokate, Pavel Krejčí. Optimal control of ODE systems involving a rate independent variational inequality. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 331-348. doi: 10.3934/dcdsb.2013.18.331 [16] Hermann Gross, Sebastian Heidenreich, Mark-Alexander Henn, Markus Bär, Andreas Rathsfeld. Modeling aspects to improve the solution of the inverse problem in scatterometry. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 497-519. doi: 10.3934/dcdss.2015.8.497 [17] T. A. Shaposhnikova, M. N. Zubova. Homogenization problem for a parabolic variational inequality with constraints on subsets situated on the boundary of the domain. Networks & Heterogeneous Media, 2008, 3 (3) : 675-689. doi: 10.3934/nhm.2008.3.675 [18] Jianlin Jiang, Shun Zhang, Su Zhang, Jie Wen. A variational inequality approach for constrained multifacility Weber problem under gauge. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1085-1104. doi: 10.3934/jimo.2017091 [19] Yekini Shehu, Olaniyi Iyiola. On a modified extragradient method for variational inequality problem with application to industrial electricity production. Journal of Industrial & Management Optimization, 2019, 15 (1) : 319-342. doi: 10.3934/jimo.2018045 [20] Valter Pohjola. An inverse problem for the magnetic Schrödinger operator on a half space with partial data. Inverse Problems & Imaging, 2014, 8 (4) : 1169-1189. doi: 10.3934/ipi.2014.8.1169

2018 Impact Factor: 1.008