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Nonlinear Quasi-hemivariational Inequalities: : Existence and Optimal Control

Authors:

Stanisław Migórski,

Akhtar A. KhanAuthors Info & Claims

SIAM Journal on Control and Optimization, Volume 59, Issue 2

Pages 1246 - 1274

https://doi.org/10.1137/19M1282210

Published: 01 January 2021 Publication History

Abstract

In this paper, we investigate a generalized nonlinear quasi-hemivariational inequality (QHI) involving a multivalued map in a Banach space. Under general assumptions, by using a fixed point theorem combined with the theory of nonsmooth analysis and the Minty technique, we prove that the set of solutions for the hemivariational inequality associated to the QHI problem is nonempty, bounded, closed, and convex. Then, we prove the existence of a solution to QHI. Furthermore, an optimal control problem governed by QVI is introduced, and a solvability result for the optimal control problem is established. Finally, an approximation of an elastic contact problem with the constitutive law involving a convex subdifferential inclusion is studied as an illustrative application, in which approximate contact boundary conditions are described by a multivalued version of the normal compliance contact condition with frictionless effect and a frictional contact law with the slip dependent coefficient of friction.

References

[1]

B. Alleche and V. D. Rădulescu, Set-valued equilibrium problems with applications to Browder variational inclusions and to fixed point theory, Nonlinear Anal. Real World Appl., 28 (2016), pp. 251--268.

[2]

D. Aussel, R. Gupta, and A. Mehra, Gap functions and error bounds for inverse quasi-variational inequality problems, J. Math. Anal. Appl. 407 (2013), pp. 270--280.

[3]

D. Aussel, A. Sultana, and V. Vetrivel, On the existence of projected solutions of quasi-variational inequalities and generalized Nash equilibrium problems, J. Optim. Theory Appl. 170 (2016), pp. 818--837.

[4]

C. Baiocchi and A. Capelo, Variational and Quasi-Variational Inequalities: Applications to Free-Boundary Problems, John Wiley, Chichester, UK, 1984.

[5]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.

[6]

A. Capatina, Variational Inequalities and Frictional Contact Problems, Adv. Mech. Math. 31, Springer, New York, 2014.

[7]

F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley Interscience, New York, 1983.

[8]

C. Clason and T. Valkonen, Primal-dual extragradient methods for nonlinear nonsmooth PDE-constrained optimization, SIAM J. Optim. 27 (2017), 1314--1339.

[9]

C. Clason, C. Tameling, and B. Wirth, Vector-valued multibang control of differential equations, SIAM J. Optim. 56 (2018), pp. 2295--2326.

[10]

C. Cornesch, T.-V. Hoarau-Mantel, and M. Sofonea, A quasistatic contact problem with slip dependent coefficient of friction for elastic materials, J. Appl. Anal. 8 (2002), pp. 63--82.

[11]

F. Demengel and P. Suquet, On locking materials, Acta Appl. Math. 6 (1986), pp. 185--211.

[12]

Z. Denkowski, S. Migórski, and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory, Kluwer, Dordrecht, 2003.

[13]

Z. Denkowski, S. Migórski, and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Applications, Kluwer, Dordrecht, 2003.

[14]

C. Eck and J. Jarusek, Existence results for the static contact problem with Coulomb friction, Math. Models Methods Appl. Sci. 8 (1998), pp. 445--468.

[15]

I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Classics in Appl. Math. 28, SIAM, Philadelphia, 1999.

[16]

H. Gfrerer and B. S. Mordukhovich, Second-order variational analysis of parametric constraint and variational systems, SIAM J. Optim. 29 (2019), pp. 423--453.

[17]

R. Glowinski, J.-L. Lions and R. Trémolières, Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, 1981.

[18]

A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.

[19]

J. Gwinner, An optimization approach to parameter identification in variational inequalities of second kind, Optim. Lett. 12 (2018), pp. 1141--1154.

[20]

J. Gwinner, B. Jadamba, A. A. Khan, and M. Sama, Identification in variational and quasi-variational inequalities, J. Convex Anal. 25 (2018), pp. 545--569.

[21]

W. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, Stud. Adv. Math. 30, AMS, Providence, RI, 2002.

[22]

M. Hintermueller, V. A. Kovtunenko, and K. Kunisch, Obstacle problems with cohesion: A hemi-variational inequality approach and its efficient numerical solution, SIAM J. Optim. 21 (2011), pp. 491--516.

[23]

M. Kamenskii, V. Obukhovskii, and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Space, Walter de Gruyter, Berlin, 2001.

[24]

A. A. Khan, S. Migórski, and M. Sama, Inverse problems for multi-valued quasi-variational inequalities and noncoercive variational inequalities with noisy data, Optimization, 68 (2019), pp. 1897--1931.

[25]

A. A. Khan and D. Motreanu, Existence theorems for elliptic and evolutionary variational and quasi-variational inequalities, J. Optim. Theory Appl. 167 (2015), pp. 1136--1161.

[26]

A. A. Khan and D. Motreanu, Inverse problems for quasi-variational inequalities, J. Global Optim. 70 (2018), pp. 401--411.

[27]

A. A. Khan and M. Sama, Optimal control of multi-valued quasi-variational inequalities, Nonlinear Anal. 75 (2012), pp. 1419--1428.

[28]

A. A. Khan, C. Tammer, and C. Zalinescu, Regularization of quasi-variational inequalities, Optimization 64 (2015), pp. 1703--1724.

[29]

N. Kikuchi and J. T. Oden, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, Stud. Appl. Math. 8, SIAM, Philadelphia, 1988.

[30]

R. Kluge, On some parameter determination problems and quasi-variational inequalities, in Theory of Nonlinear Operators, Vol. 6, Akademie-Verlag, Berlin, 1978, pp. 129--139.

[31]

V. A. Kovtunenko and K. Ohtsuka, Shape differentiability of Lagrangians and application to Stokes problem, SIAM J. Control Optim. 56 (2018), pp. 3668--3684.

[32]

Z. H. Liu, Existence results for quasilinear parabolic hemivariational inequalities, J. Differential Equations 244 (2008), pp. 1395--1409.

[33]

Z. H. Liu, S. Migórski, and S. D. Zeng, Partial differential variational inequalities involving nonlocal boundary conditions in Banach spaces, J. Differential Equations 263 (2017), pp. 3989--4006.

[34]

Z. H. Liu, D. Motreanu, and S. D. Zeng, Nonlinear evolutionary systems driven by quasi-hemivariational inequalities, Math. Methods Appl. Sci. 41 (2018), pp. 1214--1229.

[35]

Z. H. Liu, D. Motreanu, and S. D. Zeng, On the well-posedness of differential mixed quasi-variational inequalities, Topol. Methods Nonlinear Anal. 51 (2018), pp. 135--150.

[36]

Z. H. Liu and B. Zeng, Optimal control of generalized quasi-variational hemivariational inequalities and its applications, Appl. Math. Optim. 72 (2015), pp. 305--323.

[37]

Z. H. Liu, and B. Zeng, Existence results for a class of hemivariational inequalities involving the stable $(g, f,\alpha)$-quasimonotonicity, Topol. Methods Nonlinear Anal. 47 (2016), pp. 195--217.

[38]

Z. H. Liu, S. D. Zeng, and D. Motreanu, Partial differential hemivariational inequalities, Adv. Nonlinear Anal. 7 (2018), pp. 571--586.

[39]

S. Migórski, A. A. Khan, and S. D. Zeng, Inverse problems for nonlinear quasi-variational inequalities with an application to implicit obstacle problems of $p$-Laplacian type, Inverse Problem 35 (2019), 035004.

[40]

S. Migórski, A. Ochal, and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Adv. Mech. Math. 26, Springer, New York, 2013.

[41]

S. Migorski, A. Ochal, and M. Sofonea, A class of variational-hemivariational inequalities in reflexive Banach spaces, J. Elasticity 127 (2017), pp. 151--178.

[42]

S. Migorski and J. Ogorzaly, A variational-hemivariational inequality in contact problem for locking materials and nonmonotone slip dependent friction, Acta Math. Sci. 37 (2017), pp. 1639--1652.

[43]

S. Migórski and S. D. Zeng, Hyperbolic hemivariational inequalities controlled by evolution equations with application to adhesive contact model, Nonlinear Anal. Real World Appl. 43 (2018), pp. 121--143.

[44]

B. S. Mordukhovich, Variational analysis of evolution inclusions, SIAM J. Optim. 18 (2007), pp. 752--777.

[45]

Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Marcel Dekker, New York, 1995.

[46]

P. D. Panagiotopoulos, Nonconvex energy functions, hemivariational inequalities and substationary principles, Acta Mech. 42 (1983), pp. 160--183.

[47]

P. D. Panagiotopoulos, Hemivariational Inequalities, Applications in Mechanics and Engineering, Springer-Verlag, Berlin, 1993.

[48]

W. Prager, On ideal-locking materials, Trans. Soc. Rheol. 1 (1957), pp. 169--175.

[49]

W. Prager, Elastic solids of limited compressibility, in Proceedings of the 9th International Congress of Applied Mechanics, Vol. 5, Brussels, 1958, pp. 205--211.

[50]

W. Prager, On elastic, perfectly locking materials, in H. Görtler, ed., Proceedings of the 11th International Congress of Applied Mechanics, Munich, 1964, Springer-Verlag, Berlin, 1966, 538--544.

[51]

M. Shillor, M. Sofonea, and J. J. Telega, Models and Analysis of Quasistatic Contact, Lecture Notes in Phys. 655, Springer, Berlin, 2004.

[52]

M. Sofonea, A nonsmooth static frictionless contact problem with locking materials, Anal. Appl., 16 (2018), pp. 851--874.

[53]

M. Sofonea and El-H. Essoufi, A piezoelectric contact problem with slip dependent coefficient of friction, Math. Model. Anal. 9 (2004), pp. 229--242.

[54]

M. Sofonea and A. Matei, Mathematical Models in Contact Mechanics, London Math. Soc. Lecture Notes, Cambridge University Press, Cambridge, UK, 2012.

[55]

M. Sofonea and S. Migórski, Variational-Hemivariational Inequalities with Applications, Monogr. Res. Notes Math., Chapman & Hall/CRC, Boca Raton, FL, 2017.

[56]

G. J. Tang and N. J. Huang, Existence theorems of the variational-hemivariational inequalities, J. Global Optim. 56 (2013), pp. 605--622.

[57]

E. Tarafdar, A fixed point theorem equivalent to the Fan-Knaster-Kuratowski-Mazurkiewicz theorem, J. Math. Anal. Appl. 128 (1987), pp. 475--479

[58]

E. Zeidler, Nonlinear Functional Analysis and Applications II A/B, Springer, New York, 1990.

[59]

Y. L. Zhang and Y.R. He, On stably quasimonotone hemivariational inequalities, Nonlinear Anal., 74 (2011), pp. 3324--3332.

Cited By

Jiang TCai DXiao YMigórski S(2024)Time-dependent elliptic quasi-variational-hemivariational inequalities: well-posedness and applicationJournal of Global Optimization10.1007/s10898-023-01324-688:2(509-530)Online publication date: 1-Feb-2024
https://dl.acm.org/doi/10.1007/s10898-023-01324-6
Jing ZYuan ZLiu ZMigórski S(2024)Optimal Control of a New Class of Parabolic Quasi Variational–Hemivariational InequalityApplied Mathematics and Optimization10.1007/s00245-024-10190-x90:2Online publication date: 14-Oct-2024
https://dl.acm.org/doi/10.1007/s00245-024-10190-x
Peng ZYang GLiu ZMigórski S(2024)Evolutionary Quasi-variational Hemivariational Inequalities: Existence and Parameter IdentificationApplied Mathematics and Optimization10.1007/s00245-023-10100-789:2Online publication date: 1-Apr-2024
https://dl.acm.org/doi/10.1007/s00245-023-10100-7
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Index Terms

Nonlinear Quasi-hemivariational Inequalities: Existence and Optimal Control
1. Mathematics of computing
  1. Mathematical analysis
2. Theory of computation
  1. Design and analysis of algorithms
    1. Mathematical optimization

Index terms have been assigned to the content through auto-classification.

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cover image SIAM Journal on Control and Optimization

SIAM Journal on Control and Optimization Volume 59, Issue 2

DOI:10.1137/sjcodc.59.2

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© 2021, Society for Industrial and Applied Mathematics.

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Society for Industrial and Applied Mathematics

United States

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Published: 01 January 2021

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Cited By

Jiang TCai DXiao YMigórski S(2024)Time-dependent elliptic quasi-variational-hemivariational inequalities: well-posedness and applicationJournal of Global Optimization10.1007/s10898-023-01324-688:2(509-530)Online publication date: 1-Feb-2024
https://dl.acm.org/doi/10.1007/s10898-023-01324-6
Jing ZYuan ZLiu ZMigórski S(2024)Optimal Control of a New Class of Parabolic Quasi Variational–Hemivariational InequalityApplied Mathematics and Optimization10.1007/s00245-024-10190-x90:2Online publication date: 14-Oct-2024
https://dl.acm.org/doi/10.1007/s00245-024-10190-x
Peng ZYang GLiu ZMigórski S(2024)Evolutionary Quasi-variational Hemivariational Inequalities: Existence and Parameter IdentificationApplied Mathematics and Optimization10.1007/s00245-023-10100-789:2Online publication date: 1-Apr-2024
https://dl.acm.org/doi/10.1007/s00245-023-10100-7
Van Anh NVan Thuy T(2024)Long-time behavior of delay differential quasi-variational–hemivariational inequalities and application to contact problemsZeitschrift für Angewandte Mathematik und Physik (ZAMP)10.1007/s00033-024-02202-175:2Online publication date: 1-Apr-2024
https://dl.acm.org/doi/10.1007/s00033-024-02202-1
Zhang XLu YLiu D(2023)Stability and Optimal Controls for Time-space Fractional Ginzburg–Landau SystemsJournal of Optimization Theory and Applications10.1007/s10957-023-02315-z199:3(1106-1129)Online publication date: 1-Dec-2023
https://dl.acm.org/doi/10.1007/s10957-023-02315-z
Liu CYu CGong ZCheong HTeo K(2023)Numerical Computation of Optimal Control Problems with Atangana–Baleanu Fractional DerivativesJournal of Optimization Theory and Applications10.1007/s10957-023-02212-5197:2(798-816)Online publication date: 8-Apr-2023
https://dl.acm.org/doi/10.1007/s10957-023-02212-5
Kovtunenko VKunisch K(2022)Shape Derivative for Penalty-Constrained Nonsmooth–Nonconvex Optimization: Cohesive Crack ProblemJournal of Optimization Theory and Applications10.1007/s10957-022-02041-y194:2(597-635)Online publication date: 1-Aug-2022
https://dl.acm.org/doi/10.1007/s10957-022-02041-y
Cen JHaddad TNguyen VZeng S(2022)Simultaneous distributed-boundary optimal control problems driven by nonlinear complementarity systemsJournal of Global Optimization10.1007/s10898-022-01155-x84:3(783-805)Online publication date: 1-Nov-2022
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Zeng SMigórski STarzia DZou LNguyen V(2022)A class of elliptic mixed boundary value problems with (p, q)-Laplacian: existence, comparison and optimal controlZeitschrift für Angewandte Mathematik und Physik (ZAMP)10.1007/s00033-022-01789-773:4Online publication date: 1-Aug-2022
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