Exponential Stabilization of an Euler-Bernoulli Beam by a Heat Equation Involving Memory Effects

Abstract

This study delves into the dynamic behavior of a coupled system comprising an Euler-Bernoulli beam equation and a heat equation with memory, governed by different heat conduction laws. These laws, represented by a parameter $m$ include the Coleman-Gurtin and Gurtin-Pipkin laws, which uniquely influence the material’s temperature evolution. By formulating a set of partial differential equations along with appropriate boundary and transmission conditions, we lay the groundwork for analyzing the coupled system’s behavior. Our investigation primarily centers on understanding the asymptotic properties of the solution semigroup within the framework of Dafermos history space. We establish exponential stability under both the Coleman-Gurtin and Gurtin-Pipkin heat conduction models by imposing distinct conditions on the relaxation function. The article concludes with a comparison of the two heat conduction laws and a discussion on potential directions for extending this research.

Keywords

beam and heat interactions memory effect by Dafermos Coleman-Gurtin law Gurtin-Pipkin law exponential stability

Get full access to this article

View all access options for this article.

References

Akil

(2022). Stability of piezoelectric beam with magnetic effect under (Coleman or Pipkin)–Gurtin thermal law. Zeitschrift fur Angewandte Mathematik und Physik , 73, 236.

Akil

Badawi

Wehbe

(2021). Stability results of a singular local interaction elastic/viscoelastic coupled wave equations with time delay. Communications on Pure and Applied Analysis, 20, 2991–3028.

Akil

Fragnelli

Issa

(2024a). The energy decay rate of a transmission system governed by the degenerate wave equation with drift and under heat conduction with the memory effect. Mathematische Nachrichten, 297(10), 3766–3796.

Akil

Nicaise

Özer

A. Ö.

Régnier

(2024b). Stability results for novel serially-connected magnetizable piezoelectric and elastic smart-system designs. Applied Mathematics and Optimization, 89, 64.

Ames

Payne

(1991). Stabilizing solutions of the equations of dynamical linear thermoelasticity backward in time. Stability and Applied Analysis of Continuous Media, 1, 243–260.

Arendt

Batty

C. J.

(1988). Tauberian theorems and stability of one-parameter semigroups. Transactions of the American Mathematical Society, 306, 837–852.

Castille

Isabelle

Claude

(2010). Longitudinal vibration mode of piezoelectric thick-film cantilever-based sensors in liquid media. Applied Physics Letters, 96, 154102.

Chepyzhov

Pata

(2006). Some remarks on stability of semigroups arising from linear viscoelasticity. Asymptotic Analysis, 46, 251–273.

Christov

Jordan

(2005). Heat conduction paradox involving second-sound propagation in moving media. Physical Review Letters, 94, 154301.

10.

Coleman

B. D.

Gurtin

M. E.

(1967). Equipresence and constitutive equations for rigid heat conductors. Zeitschrift für angewandte Mathematik und Physik ZAMP, 18, 199–208.

11.

Dafermos

(1970). Asymptotic stability in viscoelasticity. Archive for Rational Mechanics and Analysis, 37, 297–308.

12.

Dahiya

Valle

(2012). Robotic tactile sensing. In Technologies and System 248 (pages XX). Springer Dordrecht.

13.

De Teresa

Zuazua

(2000). Null controllability of linear and semilinear heat equations in thin domains. Asymptotic Analysis, 24, 295–317.

14.

Dell’Oro

(2021). On the stability of bresse and timoshenko systems with hyperbolic heat conduction. Journal of Differential Equations, 281, 148–198.

15.

Dell’Oro

Jorge da Silva

Pinheiro

(2022). Exponential stability of Timoshenko-Gurtin-Pipkin systems with full thermal coupling. Discrete and Continuous Dynamical Systems - S, 15, 2189–2207.

16.

Dell’Oro

Pata

(2023). A hierarchy of heat conduction laws. Discrete and Continuous Dynamical Systems - S, 16, 2636–2648.

17.

Dell’Oro

Paunonen

Seifert

(2023). Optimal decay for a wave-heat system with Coleman–Gurtin thermal law. Journal of Mathematical Analysis and Applications, 518(2), 126706.

18.

Elbadry

M. M.

Ghali

(1983a). Nonlinear-temperature distribution and its effects on bridges. IABSE proceedings, 7, 169–191.

19.

Elbadry

M. M.

Ghali

(1983b). Temperature variations in concrete bridges. Journal of Structural Engineering (United States), 109, 2355–2374.

20.

Eslami

M. R.

Kiani

(2014). Slender beams, thermal buckling. In Encyclopedia of Thermal Stresses (pp.4427–4432). Dordrecht, Springer Netherlands.

21.

Fichera

(1992). Is the fourier theory of heat propagation paradoxical? Rendiconti del Circolo Matematico di Palermo, 41, 5–28.

22.

Fourier

(1988). Théorie analytique de la chaleur. Reprint of the 1822 original. Editions Jacques Gabayl:Paris.

23.

Freitas

Ramos

Özer

Almeida Júnior

(2021). Long-time dynamics for a fractional piezoelectric system with magnetic effects and Fourier’s law. Journal of Differential Equations, 280, 891–927.

24.

Gatti

Miranville

Pata

Zelik

(2008). Attractors for semi-linear equations of viscoelasticity with very low dissipation. Rocky Mountain Journal of Mathematics, 38(4), 1117–1138.

25.

Ghali

Favre

Elbadry

(2002). Stresses and deformations: Analysis and design for serviceability. In Concrete structures (p.608). London, CRC Press.

26.

Grasselli

Pata

(2002). Uniform attractors of nonautonomous dynamical systems with memory (pp.155–178). Birkhäuser Basel, Basel.

27.

Green

A. E.

Naghdi

P. M.

(1885). A re-examination of the basic postulates of thermomechanics. Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, 432, 171–194.

28.

G. Y.

Zhu

L. M.

C. Y.

Ding

Fatikow

(2016). Modeling and control of piezo-actuated nanopositioning stages: A survey. IEEE Transactions on Automation Science and Engineering, 13, 313–332.

29.

Gurtin

M. E.

Pipkin

A. C.

(1968). A general theory of heat conduction with finite wave speeds. Archive for Rational Mechanics and Analysis, 32, 113–126.

30.

Han

Z. J.

Zuazua

(2017). Decay rates for elastic-thermoelastic star-shaped networks. Networks and Heterogeneous Media, 12, 461–488.

31.

Hansen

S. W.

(2004). Several related models for multilayer sandwich plates. Mathematical Models & Methods in Applied Sciences, 14, 1103–1132.

32.

Huang

F. L.

(1985). Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces. Annals of Differential Equations, 1, 43–56.

33.

Jacques-Louis

Enrico

Lions

J.-L.

(1968). Problèmes aux limites non homogènes et applications. 1 Travaux et recherches mathématiques. Dunod, Paris.

34.

Kiran

M. R.

Farrok

Abdullah-Al-Mamun

Islam

M. R.

(2020). Progress in piezoelectric material based oceanic wave energy conversion technology. IEEE Access, 8, 146428.

35.

Kodur

V. K. R.

Agrawal

(2016). An approach for evaluating residual capacity of reinforced concrete beams exposed to fire. Engineering Structures, 110, 293–306.

36.

Lebeau

Zuazua

(1999). Decay rates for the three-dimensional linear system of thermoelasticity. Archive for Rational Mechanics and Analysis, 148, 179–231.

37.

Liu

Özer

A. Ö.

Wang

(2022). Longtime dynamics for a novel piezoelectric beam model with creep and thermo-viscoelastic effects. Nonlinear Analysis: Real World Applications, 68, 103666.

38.

Liu

Zhang

(2023). Exponential stability of a transmission system coupling plate equation and heat equation with memory. Journal of Mathematical Analysis and Applications, 518(2), 126707.

39.

Liu

Zheng

(1999). Semigroups associated with dissipative systems. In Chapman & Hall/CRC Research Notes in Mathematics Boca Raton, FL.

40.

Munoz Rivera

J. E.

Naso

M. G.

Vuk

(2004). Asymptotic behavior of the energy for electromagnetic systems with memory. Mathematical Methods in the Applied Sciences, 27, 819–841.

41.

Nunziato

J. W.

(1971). On heat conduction in materials with memory. Quarterly of Applied Mathematics, 29, 187–204.

42.

Özer

A. Ö.

Hansen

(2010). Exact boundary controllability of an abstract Mead-Marcus sandwich beam model. 49th IEEE Conference on Decision and Control (CDC), 2578–2583.

43.

Özer

A. Ö.

Khalilullah

(2024). Uniformly exponentially stable finite-difference model reduction of heat and piezoelectric beam interactions with static or hybrid feedback controllers. Evolution Equations and Control Theory. Early Access Available.

44.

Özer

A. Ö.

Khalilullah

Rasaq

(2024). The exponential stabilization of a heat and piezoelectric beam interaction with static or hybrid feedback controllers. In 2024 IEEE american control conference (ACC) (pp.5327–5332).

45.

Özer

A. Ö.

Morris

(2019). Modeling and stabilization of current-controlled piezoelectric beams with dynamic electromagnetic field. ESAIM: Control, Optimisation and Calculus of Variations, 24, 1–24.

46.

Prüss

(1984). On the spectrum of c0-semigroups. Transactions of the American Mathematical Society, 284, 847–857.

47.

Tebou

L. T.

Zuazua

(2007). Uniform boundary stabilization of the finite difference space discretization of the 1-d wave equation. Advances in Computational Mathematics, 26, 337–365.

48.

Vinogradov

A. M.

Schmidt

V. H.

Tuthill

G. F.

(2004). Damping and electromechanical energy losses in the piezoelectric polymer PVDF. Mechanics of Materials, 36, 1007–1016.

49.

Wang

J. M.

Krstic

(2015). Stability of an interconnected system of Euler-Bernoulli beam and heat equation with boundary coupling. ESAIM: COCV, 21, 1029–1052.

50.

Zhang

(2014). Stability analysis of an interactive system of wave equation and heat equation with memory. Zeitschrift für angewandte Mathematik und Physik, 65, 905–923.

51.

Zhang

Wang

J.-M.

Guo

B.-Z.

(2014). Stabilization of the Euler–Bernoulli equation via boundary connection with heat equation. Mathematics of Control, Signals, and Systems, 26, 77–118.

52.

Zhang

Zuazua

(2007). Long-time behavior of a coupled heat-wave system arising in fluid-structure interaction. Archive for Rational Mechanics and Analysis, 184, 49–120.