Exact controllability for nonlinear thermoviscoelastic plate problem

• Autorzy: Moulahi Taoufik , Aouadi Moncef , Althobaiti Saad

Identyfikatory

DOI

10.1515/dema-2024-0071

• Języki publikacji: EN

Abstrakty

EN

In this article, we consider a problem of exact controllability in the processes described by a nonlinear damped thermoviscoelastic plate. First, we prove the global well-posedness result for the nonlinear functions that are continuous with respect to time and globally Lipschitz with respect to space variable. Next, we perform a spectral analysis of the linear and uncontrolled problem. Then, we prove that the corresponding solutions decay exponentially to zero at a rate determined explicitly by the physical constants. Finally, we prove the exact controllability of the linear and the nonlinear problems by proving that the corresponding controllability mappings are surjective.

Słowa kluczowe

EN

nonlinear thermoviscoelastic plate; spectral analysis; optimal decay rate; exact controllability

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2024
• Tom: Vol. 57, nr 1
• Strony: art. no. 20240071
• Opis fizyczny: Bibliogr. 28 poz.

Twórcy

autor

Moulahi Taoufik

• Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Box 173 Al-Kharj 11942, Saudi Arabia

• https://orcid.org/0000-0002-2884-8455

autor

Aouadi Moncef

• Université de Carthage, Ecole Nationale d’Ingénieurs de Bizerte, BP66, 7035, Bizerte, Tunisia

• https://orcid.org/0000-0003-3400-5670

autor

Althobaiti Saad

• Department of Science and Technology, University College Ranyah, Taif University, Ranyah, 21975, Saudi Arabia

• https://orcid.org/0000-0003-2678-2066

Bibliografia

• [1] I. Lasiecka and R. Triggiani, Exact null controllability of structurally damped and thermoelastic parabolic models, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 9 (1998), no. 1, 43–69.
• [2] G. Avalos, Exact controllability of a thermo elastic system with control in the thermal component only, Differential Integral Equations 13 (2000), no. 4–6, 613–630, DOI: https://doi.org/10.57262/die/1356061241.
• [3] A. Benabdallah and M. G. Naso, Null controllability of a thermo elastic plate, Abstr. Appl. Anal. 7 (2002), no. 11, 585–599, DOI: https://doi.org/10.1155/S108533750220408X.
• [4] G. Lebeau and L. Robbiano, Exact control of the heat equation, Comm. Partial Differential Equations 20 (1995), no. 1–2, 335–356, DOI: https://doi.org/10.1080/03605309508821097.
• [5] L. Teresa and E. Zuazua, Controllability of the linearsystem of thermoelastic plates, Adv. Differential Equations 1 (1996), no. 3, 369–402, DOI: https://doi.org/10.57262/ade/1366896044.
• [6] H. Leiva and H. Zambrano, Rank condition for the controllability of a linear time-varying system, Internat. J. Control 72 (1999), no. 10, 929–931, DOI: https://doi.org/10.1080/002071799220669.
• [7] S. K. Chang and R. Triggiani, Spectral Analysis of Thermoelastic Plates, Optimal Control: Theory, Algorithms and Applications, Springer, Kluwer, 1998.
• [8] R. Triggiani and J. Zhang, Heat-viscoelastic plate interaction: Analyticity, spectral analysis, exponential decay, Evol. Equ. Control Theory 7 (2018), no. 1, 153–182, DOI: https://doi.org/10.3934/eect.2018008.
• [9] I. Lasiecka and T. Seidman, Blowup estimates for observability of a thermoelastic system, Asymptot. Anal. 50 (2006), no. 1–2, 93–120, https://content.iospress.com/articles/asymptotic-analysis/asy789.
• [10] C. Castro and L. Teresa, Null controllability of the linear system of thermoelastic plates, Aust. J. Math. Anal. Appl. 428 (2015), no. 2, 772–793, DOI: https://doi.org/10.1016/j.jmaa.2015.03.014.
• [11] S. Hansen and B. Y. Zhang, Boundary controllability of a linear thermoelastic beam, Aust. J. Math. Anal. Appl. 210 (1997), no. 1, 282–205, DOI: https://doi.org/10.1006/jmaa.1997.5437.
• [12] M. Eller, I. Lasiecka, and R. Triggiani, Simultaneous exact/approximate boundary controllability of thermo-elastic plates with variable thermal coefficient and moment control, Aust. J. Math. Anal. Appl. 251 (2000), no. 2, 452–478, DOI: https://doi.org/10.1006/jmaa.2000.7015.
• [13] M. Eller, I. Lasiecka, and R. Triggiani, Exact/approximate controllability of thermoelastic plates with variable thermal coefficients, Discrete Contin. Dyn. Syst. 7 (2001), no. 2, 283–302, DOI: https://doi.org/10.3934/dcds.2001.7.283.
• [14] F. Yang and P. Yao, Boundary controllability of a thermo elastic plate, 2016 35th Chinese Control Conference (CCC), 2016, pp. 1317–1322, DOI: https://doi.org/10.1109/chicc.2016.7553271.
• [15] G. Avalos and I. Lasiecka, The null controllability of thermoelastic plates and singularity of the associated minimal energy function, Aust. J. Math. Anal. Appl. 294 (2004), no. 1, 34–61, DOI: https://doi.org/10.1016/j.jmaa.2004.01.035.
• [16] H. Leiva, Controllability of the strongly damped wave equation with impulsesand delay, Nonauton. Dyn. Syst. 4 (2017), no. 1, 31–39, DOI: https://doi.org/10.1515/msds-2017-0004.
• [17] H. Leiva, Exact controllability of semilinear evolution equation and applications, Int. J. Syst. Control Commun. 1 (2008), no. 1, 1–12, DOI: https://doi.org/10.1504/IJSCC.2008.019580.
• [18] H. Leiva, Exact controllability of the suspension bridge model proposed by Lazer and McKenna, Aust. J. Math. Anal. Appl. 309 (2005), no. 2, 404–419, DOI: https://doi.org/10.1016/j.jmaa.2004.07.025.
• [19] A. Carrasco, H. Leiva, and J. Uzcátegui, Controllability of Semilinear Evolution Equations, Caracas, Venezuela, ISBN 978-980-261-162-1, 2015.
• [20] R. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, vol. 21, Springer, New York, 1995.
• [21] M. Aouadi and T. Moulahi, The controllability of a thermo elastic plate problem revisited, Evol. Equ. Control Theory 7 (2018), no. 2, 1–31, DOI: https://doi.org/10.3934/eect.2018001.
• [22] R. Curtain and A. Pritchard, Infinite Dimensional Linear Systems, Lecture Notes in Control and Information Sciences, Springer, Berlin, 1978.
• [23] M. Aouadi, I. Mahfoudhi, and T. Moulahi, Boundary stabilization of a thermoelastic diffusion system of type II, Acta Appl. Math. 169 (2020), no. 2, 1–24, DOI: https://doi.org/10.1007/s10440-019-00308-7.
• [24] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Science & Business Media, Berlin, 1983.
• [25] W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in Fortran 77, Citeseer by the Press Syndicate of the University, Cambridge, 1997.
• [26] D. Ya. Khusainov and M. Pokojovy, Solving the linear 1D thermoelasticity equations with pure delay, Int. J. Math. Sci. 2015 (2015), 1–12, DOI: https://doi.org/10.1155/2015/479267.
• [27] J. Zabczyk, Mathematical Control Theory: An Introduction, Reprint of the 1995 edition, Birkhäuser, USA, 1995.
• [28] M. Aouadi and T. Moulahi, Approximate controllability of abstract nonsimple thermoelastic problem, Evol. Equ. Control Theory 4 (2015), no. 2, 373–389, DOI: https://doi.org/10.3934/eect.2015.4.373.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr POPUL/SP/0154/2024/02 w ramach programu "Społeczna odpowiedzialność nauki II" - moduł: Popularyzacja nauki (2026).