The present paper is inspired by the work of Bidouan et al, Setting Of Some Criteria for Piezoelectric Materials Stabilizability, Journal of Current Engineering and Technology, Volume 5, Issue 1, 2023, in which the authors have identified 11 types of piezoelectric materials that can be stabilized by utilizing an electromagnetic field. However, Bidouan et al. established stabilization for a steady state as the target state. Here, we succeed in stabilizing the classical solution of the piezoelectric equations around an evolutionary state. The control is enriched by a magnetic regularizing term to achieve H1 regularity for the electromagnetic field, which solves a trace problem encountered in Bidouan et al. Furthermore, the decay rate of the perturbation energy is significantly improved by the regularizing term. The internal control implemented in this work is purely magnetic. As for the existence of the results of the controlled solutions, the Faedo-Galerkin method is used.
| Published in | American Journal of Applied Mathematics (Volume 13, Issue 4) |
| DOI | 10.11648/j.ajam.20251304.16 |
| Page(s) | 292-307 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2025. Published by Science Publishing Group |
Control Theory, Partial Differential Equations, Piezoelectricity
| [1] | Banks, H. T., Smith, R. C., and Wang, Y., Smart material structures-modeling, estimation and control, Chichester, United Kingdom and New York/Paris: John Wiley & Sons/Masson, 1996. |
| [2] | Bernadou M. and Haenel C., Numerical analysis of piezoelectric shells, Plates and Shells, M. Fortin, ed., CRM Proc. Lecture Notes, Amer. Math. Soc., Providence, Vol. 21, RI, pp. 55-63, 1999. |
| [3] | Bidouan, R., Sène, A., Marcos, A. Setting of some criteria for piezoelectric materials stabilizability, Journal of Current Engineering and Technology, 5(1). ISSN: 2582-1210. |
| [4] | Destuynder Ph., Legrain I., Castel L. and Richard N., Theoretical, numerical and experimental discussion on the use of piezoelectric devices for control-structure interaction, European J. Mech. Solids, 181-213, 11(1992). |
| [5] | Duvant, G.andLions, J. L., Inequalities in mechanicsand physics. Springer Science & Business Media, vol. 219, 2012. |
| [6] | Cady, W. G., Piezoelectricity: An introduction to the theory and applications of electromechancial phenomena in crystals, McGraw-Hill Book Company, Incorporated, 1946. |
| [7] | Hansen, S., Analysis of a plate with a localized piezoelectric patch, In Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No. 98CH36171), vol. 3, IEEE, pp. 2952-2957, (1998). |
| [8] | Ikeda, T., Fundamentals of piezoelectricity, Oxford university press, 1996. |
| [9] | Raoult, A., and Sène, A. Modelling of piezoelectric plates including magnetic effects, Asymptotic Analysis 34, 1-40, 1 (2003), |
| [10] | Sène, A., Modélisation asymptotique de plaques: Contrôlabilité exacte frontière, piézoélectricité, PhD thesis, Université Joseph-Fourier-Grenoble I, 1999. |
| [11] | Smith, R., Smart material systems, Society for Industrial and Applied Mathematics, jan. 2005, |
| [12] | Taylor G. W., Gagepain J. J., Meeker T. R., Nakamura T. and Shuvalov L. A., “Piezolelectricity”, Vol. 4, 1985, |
| [13] | Tiersten, H. F., Linear piezoelectric plate vibrations: Elements of the linear theory of piezoelectricity and the vibrations piezoelectric plates, Springer, 2013. |
| [14] | Wang, J., and Yang, J. S. (2000), Higher-order theories of piezoelectric plates and applications, Applied. Mechanics. Reviews, 53, 87-99, |
| [15] | Yang, J. S., and Hu, Y. T. (2004), Mechanics of electroelastic bodies under biasing fields, Applied. Mechanics. Reviews., pp. 173-189 57, |
| [16] | Yang, J., et al., An introduction to the theory of piezoelectricity, vol. 9. Springer, 2005. |
| [17] | Yang J. (2006), A review of a few topics in piezoelectricity, Applied. Mechanics. Reviews., (59), pp. 335-345, |
APA Style
Bidouan, R., Sène, A., Marcos, A. (2025). Stabilization of Piezoelectric Body Deformations Around an Arbitrary Trajectory via an Electromagnetic Field. American Journal of Applied Mathematics, 13(4), 292-307. https://doi.org/10.11648/j.ajam.20251304.16
ACS Style
Bidouan, R.; Sène, A.; Marcos, A. Stabilization of Piezoelectric Body Deformations Around an Arbitrary Trajectory via an Electromagnetic Field. Am. J. Appl. Math. 2025, 13(4), 292-307. doi: 10.11648/j.ajam.20251304.16
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Download@article{10.11648/j.ajam.20251304.16,
author = {Romziath Bidouan and Abdou Sène and Aboubacar Marcos},
title = {Stabilization of Piezoelectric Body Deformations Around an Arbitrary Trajectory via an Electromagnetic Field
},
journal = {American Journal of Applied Mathematics},
volume = {13},
number = {4},
pages = {292-307},
doi = {10.11648/j.ajam.20251304.16},
url = {https://doi.org/10.11648/j.ajam.20251304.16},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20251304.16},
abstract = {The present paper is inspired by the work of Bidouan et al, Setting Of Some Criteria for Piezoelectric Materials Stabilizability, Journal of Current Engineering and Technology, Volume 5, Issue 1, 2023, in which the authors have identified 11 types of piezoelectric materials that can be stabilized by utilizing an electromagnetic field. However, Bidouan et al. established stabilization for a steady state as the target state. Here, we succeed in stabilizing the classical solution of the piezoelectric equations around an evolutionary state. The control is enriched by a magnetic regularizing term to achieve H1 regularity for the electromagnetic field, which solves a trace problem encountered in Bidouan et al. Furthermore, the decay rate of the perturbation energy is significantly improved by the regularizing term. The internal control implemented in this work is purely magnetic. As for the existence of the results of the controlled solutions, the Faedo-Galerkin method is used.
},
year = {2025}
}
TY - JOUR T1 - Stabilization of Piezoelectric Body Deformations Around an Arbitrary Trajectory via an Electromagnetic Field AU - Romziath Bidouan AU - Abdou Sène AU - Aboubacar Marcos Y1 - 2025/08/20 PY - 2025 N1 - https://doi.org/10.11648/j.ajam.20251304.16 DO - 10.11648/j.ajam.20251304.16 T2 - American Journal of Applied Mathematics JF - American Journal of Applied Mathematics JO - American Journal of Applied Mathematics SP - 292 EP - 307 PB - Science Publishing Group SN - 2330-006X UR - https://doi.org/10.11648/j.ajam.20251304.16 AB - The present paper is inspired by the work of Bidouan et al, Setting Of Some Criteria for Piezoelectric Materials Stabilizability, Journal of Current Engineering and Technology, Volume 5, Issue 1, 2023, in which the authors have identified 11 types of piezoelectric materials that can be stabilized by utilizing an electromagnetic field. However, Bidouan et al. established stabilization for a steady state as the target state. Here, we succeed in stabilizing the classical solution of the piezoelectric equations around an evolutionary state. The control is enriched by a magnetic regularizing term to achieve H1 regularity for the electromagnetic field, which solves a trace problem encountered in Bidouan et al. Furthermore, the decay rate of the perturbation energy is significantly improved by the regularizing term. The internal control implemented in this work is purely magnetic. As for the existence of the results of the controlled solutions, the Faedo-Galerkin method is used. VL - 13 IS - 4 ER -
Department of Fundamental and Applied Mathematics, Institute of Mathematics and Physical Sciences, Abomey-Calavi University, Porto-Novo, Benin
Department of Applied Mathematics and Computer Science, Pole of Science, Technology and Digital, Cheikh Hamidou Kane Digital University, Dakar, Senegal
Department of Fundamental and Applied Mathematics, Institute of Mathematics and Physical Sciences, Abomey-Calavi University, Porto-Novo, Benin
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