Abstract
This paper focuses on optimal control of a coupled system modeling a dynamic frictional thermo-electroviscoelastic contact problem between a piezoelectric body and an electrically and thermally conductive foundation. The material’s behavior is described by a linear thermo-viscoelectroelastic constitutive law and the contact is modeled with a compliance normal condition coupled with Coulomb’s friction law, an electric condition in the Tresca’s form and thermal condition taking into account both heat transfer and frictional heating process. The weak formulation of the problem consists of a system of two variational inequalities and nonlinear variational equations. We provide existence and uniqueness results of a weak solution to the model and, under some additional assumptions, the continuous dependence of a solution on the problem’s data. Finally, for a class of optimal control problems and inverse problems, we prove the existence of optimal solutions.
Similar content being viewed by others
References
Amassad, A., Kuttler, K.L., Rochdi, M., Shillor, M.: Quasi-static thermoviscoelastic contact problem with slip dependent friction coefficient. Math. Comput. Model. 36(7–8), 839–854 (2002)
Baiz, O., Benaissa, H., El Moutawakil, D., Fakhar, R.: Variational and numerical analysis of a quasistatic thermo-electro-visco-elastic frictional contact problem. Z. Angew. Math. Mech. 99(3), 1–20 (2018)
Boukrouche, M., Tarzia, D.A.: Existence, uniqueness and convergence of optimal control problems associated with parabolic variational inequalities of the second kind. Nonlinear Anal., Real World Appl. 12, 2211–2224 (2011)
Barboteu, M., Bartosz, K., Danan, D.: Analysis of a dynamic contact problem with nonmonotone friction and nonclamped boundary conditions. Appl. Numer. Math. 126, 53–77 (2018)
Barbu, V.: Optimal Control of Variational Inequalities. Research Notes in Mathematics, vol. 100. Pitman, Boston (1984)
Bartosz, K., Danan, D., Szafraniec, P.: Numerical analysis of a dynamic bilateral thermoviscoelastic contact problem with nonmonotone friction law. Comput. Math. Appl. 73, 727–746 (2017)
Brézis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer, New York (2011)
Campo, M., Fernandez, J.R., Han, W.: A dynamic viscoelastic contact problem with normal compliance and damage. Finite Elem. Anal. Des. 42, 1–24 (2005)
Chen, T., Huang, N.J., Xiao, Y.B.: Variational and numerical analysis of a dynamic viscoelastic contact problem with friction and wear. Optimization (2020). https://doi.org/10.1080/02331934.2020.1712394
Denkowski, Z., Migorski, S., Ochal, A.: A class of optimal control problems for piezoelectric frictional contact models. Nonlinear Anal., Real World Appl. 12, 1883–1895 (2011)
Freidman, A.: Optimal control for variational inequalities. SIAM J. Control Optim. 24, 439–451 (1986)
Fremond, M.: Non-smooth Thermomechanics. Springer, Berlin (2002)
Guo, F.M., et al.: Study on low voltage actuated MEMS rf capacitive switches. Sens. Actuators A, Phys. 108, 128–133 (2003)
Han, W., Sofonea, M.: Quasistatic Contact Problems in Viscoelastictity and Viscoplasticity. Studies in Advanced Mathematics. Am. Math. Soc./International Press, Providence (2002)
Hüeber, S., Wohlmuth, B.I.: Thermo-mechanical contact problems on non-matching meshes. Comput. Methods Appl. Mech. Eng. 198(15–16), 1338–1350 (2009)
Kulig, A.: Hyperbolic hemivariational inequalities for dynamic viscoelastic contact problems. J. Elast. 110, 1–31 (2013)
Lamhamdi, M., et al.: Voltage and temperature effect on dielectric charging for RF-MEMS capacitive switches reliability investigation. Microelectron. Reliab. 48(8–9), 1248–1252 (2008)
Matei, A., Micu, S.: Boundary optimal control for a frictional contact problem with normal compliance. Appl. Math. Optim. 78, 379–401 (2018)
Matei, A., Sofonea, M.: A mixed variational formulation for a piezoelectric frictional contact problem. IMA J. Appl. Math. 82(2), 334–354 (2017)
Mignot, F., Puel, J.P.: Optimal control in some variational inequalities. SIAM J. Control Optim. 22, 466–476 (1984)
Migórski, S.: Dynamic hemivariational inequality modeling viscoelastic contact problem with normal damped response and friction. Appl. Anal. 84, 669–699 (2005)
Migórski, S., Ochal, A., Sofonea, M.: Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems. Advances in Mechanics and Mathematics, vol. 26. Springer, New York (2013)
Migórski, S., Szafraniec, P.: A class of dynamic frictional contact problems governed by a system of hemivariational inequalities in thermoviscoelasticity. Nonlinear Anal., Real World Appl. 15, 158–171 (2014)
Peng, F.: Actuator placement optimization and adaptive vibration control of plate smart structures. J. Intell. Mater. Syst. Struct. 16, 263–271 (2005)
Peng, Z., Kunisch, K.: Optimal control of elliptic variational-hemivriational inequalities. Int. J. Optim.: Theory Methods Appl. 178, 1–25 (2018)
Schauwecker, B., et al.: Investigations of rf shunt airbridge switches among different environmental conditions. Sens. Actuators A, Phys. 114, 49–58 (2004)
Rochdi, M., Shillor, M.: Existence and uniqueness for a quasistatic frictional bilateral contact problem in thermoviscoelasticity. Q. Appl. Math. 58(3), 543–560 (2000)
Shillor, M., Sofonea, M., Telega, J.J.: Models and Analysis of Quasistatic Contact. Lecture Notes in Physics, vol. 655. Springer, Berlin (2004)
Sofonea, M.: Convergence results and optimal control for a class of hemivariational inequalities. SIAM J. Math. Anal. 50, 4066–4086 (2018)
Sofonea, M., Patrulescu, F.: A viscoelastic contact problem with adhesion and surface memory effects. Math. Model. Anal. 19(5), 607–626 (2014)
Touzaline, A.: Optimal control of a frictional contact problem. Acta Math. Appl. Sin. Engl. Ser. 31, 991–1000 (2015)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Baiz, O., Benaissa, H. Optimal Control for a Dynamic Thermo-Electro-Viscoelastic Contact Problem with Frictional Heating. Acta Appl Math 171, 22 (2021). https://doi.org/10.1007/s10440-021-00390-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10440-021-00390-w
Keywords
- Linear thermo-electroelastic materials
- Dynamic frictional contact problem
- Frictional heating and heat transfer
- Variational coupled system
- Weak solution
- Continuous dependence
- Optimal control