doi: 10.3934/eect.2021064
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Continuous dependence and optimal control of a dynamic elastic-viscoplastic contact problem with non-monotone boundary conditions

School of Mathematical Science, Zhejiang University, Hangzhou 310027, China

*Corresponding author: Xilu Wang

Received  June 2021 Revised  October 2021 Early access January 2022

In this paper, we consider continuous dependence and optimal control of a dynamic elastic-viscoplastic contact model with Clarke subdifferential boundary conditions. Since the constitutive law of elastic-viscoplastic materials has an implicit expression of the stress field, the weak form of the model is an evolutionary hemivariational inequality coupled with an integral equation. By providing some equivalent weak formulations, we prove the continuous dependence of the solution on external forces and initial conditions in the weak topologies. Finally, the existence of optimal solutions to a boundary optimal control problem is established.

Citation: Xilu Wang, Xiaoliang Cheng. Continuous dependence and optimal control of a dynamic elastic-viscoplastic contact problem with non-monotone boundary conditions. Evolution Equations & Control Theory, doi: 10.3934/eect.2021064
References:
[1]

M. BarboteuA. Matei and M. Sofonea, On the behavior of the solution of a viscoplastic contact problem, Quarterly of Applied Mathematics, 72 (2014), 625-647.  doi: 10.1090/S0033-569X-2014-01345-4.  Google Scholar

[2]

A. Benraouda and M. Sofonea, A convergence result for history-dependent quasivariational inequalities, Applicable Analysis, 96 (2017), 2635-2651.  doi: 10.1080/00036811.2016.1236920.  Google Scholar

[3]

X. ChengS. MigórskiA. Ochal and M. Sofonea, Analysis of two quasistatic history-dependent contact models, Discrete and Continuous Dynamical Systems-Series B, 19 (2014), 2425-2445.  doi: 10.3934/dcdsb.2014.19.2425.  Google Scholar

[4]

X. Cheng and X. Wang, Numerical analysis of a history-dependent variational-hemivariational inequality for a viscoplastic contact problem, International Journal of Numerical Analysis and Modeling, 17 (2020), 820-838.   Google Scholar

[5]

X. ChengQ. XiaoS. Migórski and A. Ochal, Error estimate for quasistatic history-dependent contact model, Computers and Mathematics with Applications, 77 (2019), 2943-2952.  doi: 10.1016/j.camwa.2018.08.058.  Google Scholar

[6]

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

[7]

Z. DenkowskiS. Migórski and A. Ochal, Optimal control for a class of mechanical thermoviscoelastic frictional contact problems, Control and Cybernetics, 36 (2007), 611-632.   Google Scholar

[8]

Z. DenkowskiS. Migórski and A. Ochal, A class of optimal control problems for piezoelectric frictional contact models, Nonlinear Analysis: Real World Applications, 12 (2011), 1883-1895.  doi: 10.1016/j.nonrwa.2010.12.006.  Google Scholar

[9]

C. Fang and W. Han, Well-posedness and optimal control of a hemivariational inequality for nonstationary Stokes fluid flow, Discrete and Continuous Dynamical Systems, 36 (2016), 5369-5386.  doi: 10.3934/dcds.2016036.  Google Scholar

[10]

D. HanW. HanS. Migórski and J. Zhao, Convergence analysis of numerical solutions for optimal control of variational-hemivariational inequalities, Applied Mathematics Letters, 105 (2020), 106327.  doi: 10.1016/j.aml.2020.106327.  Google Scholar

[11]

J. Han and H. Zeng, Variational analysis and optimal control of dynamic unilateral contact models with friction, Journal of Mathematical Analysis and Applications, 473 (2019), 712-748.  doi: 10.1016/j.jmaa.2018.12.068.  Google Scholar

[12]

W. HanS. Migórski and M. Sofonea, A class of variational-hemivariational inequalities with applications to frictional contact problems, SIAM Journal on Mathematical Analysis, 46 (2014), 3891-3912.  doi: 10.1137/140963248.  Google Scholar

[13]

W. Han and M. Sofonea, Numerical analysis of hemivariational inequalities in contact mechanics, Acta Numerica, 28 (2019), 175-286.  doi: 10.1017/S0962492919000023.  Google Scholar

[14]

W. HanM. Sofonea and M. Barboteu, Numerical analysis of elliptic hemivariational inequalities, SIAM Journal on Numerical Analysis, 55 (2017), 640-663.  doi: 10.1137/16M1072085.  Google Scholar

[15]

C. Jiang and B. Zeng, Continuous dependence and optimal control for a class of variational-hemivariational inequalities, Applied Mathematics and Optimization, 82 (2020), 637-656.  doi: 10.1007/s00245-018-9543-4.  Google Scholar

[16]

A. Kulig, A quasistatic viscoplastic contact problem with normal compliance, unilateral constraint, memory term and friction, Nonlinear Analysis: Real World Applications, 33 (2017), 226-236.  doi: 10.1016/j.nonrwa.2016.06.007.  Google Scholar

[17]

A. Kulig, Variational-hemivariational approach to quasistatic viscoplastic contact problem with normal compliance, unilateral constraint, memory term, friction and damage, Nonlinear Analysis: Real World Applications, 44 (2018), 401-416.  doi: 10.1016/j.nonrwa.2018.05.014.  Google Scholar

[18]

Y. Li, A dynamic contact problem for elastic-viscoplastic materials with normal damped response and damage, Applicable Analysis, 95 (2016), 2485-2500.  doi: 10.1080/00036811.2015.1094797.  Google Scholar

[19]

S. Migórski, Optimal control of history-dependent evolution inclusions with applocations to frictional contact, Journal of Optimization Theory and Applications, 185 (2020), 574-596.  doi: 10.1007/s10957-020-01659-0.  Google Scholar

[20]

S. MigórskiA. Ochal and M. Sofonea, Analysis of a dynamic elastic-viscoplastic contact problem with friction, Discrete and Continuous Dynamical Systems-Series B, 10 (2008), 887-902.  doi: 10.3934/dcdsb.2008.10.887.  Google Scholar

[21]

S. MigórskiA. Ochal and M. Sofonea, Integrodifferential hemivariational inequalities with applications to viscoelastic frictional contact, Mathematical Models and Methods in Applied Sciences, 18 (2008), 271-290.  doi: 10.1142/S021820250800267X.  Google Scholar

[22]

S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, in: Advances in Mechanics and Mathematics, vol. 26, Springer, New York, 2013. doi: 10.1007/978-1-4614-4232-5.  Google Scholar

[23]

Z. Peng, P. Gamorski and S. Migórski, Boundary optimal control of a dynamic frictional contact problem, ZAMM Journal of Applied Mathematics and Mechanics: Zeitschrift fur Angewandte Mathematik und Mechanik, 100 (2020), e201900144. doi: 10.1002/zamm.201900144.  Google Scholar

[24]

M. Selmani and L. Selmani, Frictional contact problem for elastic-viscoplastic materials with thermal effect, Applied Mathematics and Mechanics (English Edition), 34 (2013), 761-776.  doi: 10.1007/s10483-013-1705-7.  Google Scholar

[25]

M. Sofonea, Convergence results and optimal control for a class of hemivariational inequalities, SIAM Journal on Mathematical Analysis, 50 (2018), 4066-4086.  doi: 10.1137/17M1144404.  Google Scholar

[26]

M. Sofonea and A. Matei, History-dependent mixed variational problems in contact mechanics, Journal of Global Optimization, 61 (2015), 591-614.  doi: 10.1007/s10898-014-0193-z.  Google Scholar

[27]

M. Sofonea and M. Shillor, A viscoplastic contact problem with a normal compliance with limited penetration condition and history-dependent stiffness coeffcient, Communications on Pure and Applied Analysis, 13 (2014), 371-387.  doi: 10.3934/cpaa.2014.13.371.  Google Scholar

[28]

M. Sofonea and Y. B. Xiao, Boundary optimal control of a nonsmooth frictionless contact problem, Computers and Mathematics with Applications, 78 (2019), 152-165.  doi: 10.1016/j.camwa.2019.02.027.  Google Scholar

[29]

M. Sofonea and Y. B. Xiao, Tykhonov well-posedness of a viscoplastic contact problem, Evolution Equations and Control Theory, 9 (2019), 1167-1185.  doi: 10.3934/eect.2020048.  Google Scholar

[30]

X. Wang and X. Cheng, Numerical analysis of a dynamic viscoplastic contact problem, International Journal of Computer Mathematics. doi: 10.1080/00207160.2021.1955107.  Google Scholar

[31]

Y. B. Xiao and M. Sofonea, On the optimal control of variational-hemivariational inequalities, Journal of Mathematical Analysis and Applications, 475 (2019), 364-384.  doi: 10.1016/j.jmaa.2019.02.046.  Google Scholar

show all references

References:
[1]

M. BarboteuA. Matei and M. Sofonea, On the behavior of the solution of a viscoplastic contact problem, Quarterly of Applied Mathematics, 72 (2014), 625-647.  doi: 10.1090/S0033-569X-2014-01345-4.  Google Scholar

[2]

A. Benraouda and M. Sofonea, A convergence result for history-dependent quasivariational inequalities, Applicable Analysis, 96 (2017), 2635-2651.  doi: 10.1080/00036811.2016.1236920.  Google Scholar

[3]

X. ChengS. MigórskiA. Ochal and M. Sofonea, Analysis of two quasistatic history-dependent contact models, Discrete and Continuous Dynamical Systems-Series B, 19 (2014), 2425-2445.  doi: 10.3934/dcdsb.2014.19.2425.  Google Scholar

[4]

X. Cheng and X. Wang, Numerical analysis of a history-dependent variational-hemivariational inequality for a viscoplastic contact problem, International Journal of Numerical Analysis and Modeling, 17 (2020), 820-838.   Google Scholar

[5]

X. ChengQ. XiaoS. Migórski and A. Ochal, Error estimate for quasistatic history-dependent contact model, Computers and Mathematics with Applications, 77 (2019), 2943-2952.  doi: 10.1016/j.camwa.2018.08.058.  Google Scholar

[6]

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

[7]

Z. DenkowskiS. Migórski and A. Ochal, Optimal control for a class of mechanical thermoviscoelastic frictional contact problems, Control and Cybernetics, 36 (2007), 611-632.   Google Scholar

[8]

Z. DenkowskiS. Migórski and A. Ochal, A class of optimal control problems for piezoelectric frictional contact models, Nonlinear Analysis: Real World Applications, 12 (2011), 1883-1895.  doi: 10.1016/j.nonrwa.2010.12.006.  Google Scholar

[9]

C. Fang and W. Han, Well-posedness and optimal control of a hemivariational inequality for nonstationary Stokes fluid flow, Discrete and Continuous Dynamical Systems, 36 (2016), 5369-5386.  doi: 10.3934/dcds.2016036.  Google Scholar

[10]

D. HanW. HanS. Migórski and J. Zhao, Convergence analysis of numerical solutions for optimal control of variational-hemivariational inequalities, Applied Mathematics Letters, 105 (2020), 106327.  doi: 10.1016/j.aml.2020.106327.  Google Scholar

[11]

J. Han and H. Zeng, Variational analysis and optimal control of dynamic unilateral contact models with friction, Journal of Mathematical Analysis and Applications, 473 (2019), 712-748.  doi: 10.1016/j.jmaa.2018.12.068.  Google Scholar

[12]

W. HanS. Migórski and M. Sofonea, A class of variational-hemivariational inequalities with applications to frictional contact problems, SIAM Journal on Mathematical Analysis, 46 (2014), 3891-3912.  doi: 10.1137/140963248.  Google Scholar

[13]

W. Han and M. Sofonea, Numerical analysis of hemivariational inequalities in contact mechanics, Acta Numerica, 28 (2019), 175-286.  doi: 10.1017/S0962492919000023.  Google Scholar

[14]

W. HanM. Sofonea and M. Barboteu, Numerical analysis of elliptic hemivariational inequalities, SIAM Journal on Numerical Analysis, 55 (2017), 640-663.  doi: 10.1137/16M1072085.  Google Scholar

[15]

C. Jiang and B. Zeng, Continuous dependence and optimal control for a class of variational-hemivariational inequalities, Applied Mathematics and Optimization, 82 (2020), 637-656.  doi: 10.1007/s00245-018-9543-4.  Google Scholar

[16]

A. Kulig, A quasistatic viscoplastic contact problem with normal compliance, unilateral constraint, memory term and friction, Nonlinear Analysis: Real World Applications, 33 (2017), 226-236.  doi: 10.1016/j.nonrwa.2016.06.007.  Google Scholar

[17]

A. Kulig, Variational-hemivariational approach to quasistatic viscoplastic contact problem with normal compliance, unilateral constraint, memory term, friction and damage, Nonlinear Analysis: Real World Applications, 44 (2018), 401-416.  doi: 10.1016/j.nonrwa.2018.05.014.  Google Scholar

[18]

Y. Li, A dynamic contact problem for elastic-viscoplastic materials with normal damped response and damage, Applicable Analysis, 95 (2016), 2485-2500.  doi: 10.1080/00036811.2015.1094797.  Google Scholar

[19]

S. Migórski, Optimal control of history-dependent evolution inclusions with applocations to frictional contact, Journal of Optimization Theory and Applications, 185 (2020), 574-596.  doi: 10.1007/s10957-020-01659-0.  Google Scholar

[20]

S. MigórskiA. Ochal and M. Sofonea, Analysis of a dynamic elastic-viscoplastic contact problem with friction, Discrete and Continuous Dynamical Systems-Series B, 10 (2008), 887-902.  doi: 10.3934/dcdsb.2008.10.887.  Google Scholar

[21]

S. MigórskiA. Ochal and M. Sofonea, Integrodifferential hemivariational inequalities with applications to viscoelastic frictional contact, Mathematical Models and Methods in Applied Sciences, 18 (2008), 271-290.  doi: 10.1142/S021820250800267X.  Google Scholar

[22]

S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, in: Advances in Mechanics and Mathematics, vol. 26, Springer, New York, 2013. doi: 10.1007/978-1-4614-4232-5.  Google Scholar

[23]

Z. Peng, P. Gamorski and S. Migórski, Boundary optimal control of a dynamic frictional contact problem, ZAMM Journal of Applied Mathematics and Mechanics: Zeitschrift fur Angewandte Mathematik und Mechanik, 100 (2020), e201900144. doi: 10.1002/zamm.201900144.  Google Scholar

[24]

M. Selmani and L. Selmani, Frictional contact problem for elastic-viscoplastic materials with thermal effect, Applied Mathematics and Mechanics (English Edition), 34 (2013), 761-776.  doi: 10.1007/s10483-013-1705-7.  Google Scholar

[25]

M. Sofonea, Convergence results and optimal control for a class of hemivariational inequalities, SIAM Journal on Mathematical Analysis, 50 (2018), 4066-4086.  doi: 10.1137/17M1144404.  Google Scholar

[26]

M. Sofonea and A. Matei, History-dependent mixed variational problems in contact mechanics, Journal of Global Optimization, 61 (2015), 591-614.  doi: 10.1007/s10898-014-0193-z.  Google Scholar

[27]

M. Sofonea and M. Shillor, A viscoplastic contact problem with a normal compliance with limited penetration condition and history-dependent stiffness coeffcient, Communications on Pure and Applied Analysis, 13 (2014), 371-387.  doi: 10.3934/cpaa.2014.13.371.  Google Scholar

[28]

M. Sofonea and Y. B. Xiao, Boundary optimal control of a nonsmooth frictionless contact problem, Computers and Mathematics with Applications, 78 (2019), 152-165.  doi: 10.1016/j.camwa.2019.02.027.  Google Scholar

[29]

M. Sofonea and Y. B. Xiao, Tykhonov well-posedness of a viscoplastic contact problem, Evolution Equations and Control Theory, 9 (2019), 1167-1185.  doi: 10.3934/eect.2020048.  Google Scholar

[30]

X. Wang and X. Cheng, Numerical analysis of a dynamic viscoplastic contact problem, International Journal of Computer Mathematics. doi: 10.1080/00207160.2021.1955107.  Google Scholar

[31]

Y. B. Xiao and M. Sofonea, On the optimal control of variational-hemivariational inequalities, Journal of Mathematical Analysis and Applications, 475 (2019), 364-384.  doi: 10.1016/j.jmaa.2019.02.046.  Google Scholar

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