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Asymptotic Behavior in a Laminated Beams Due Interfacial Slip with a Boundary Dissipation of Fractional Derivative Type

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Abstract

We consider a laminated beams due interfacial slip with control boundary conditions of fractional derivative type. We show the existence and uniqueness of solutions. Furthermore, concerning the asymptotic behavior we show the lack of exponential stability and the polynomial decay rate of the corresponding semigroup by using the classic theorem of Borichev and Tomilov.

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References

  1. Borichev, A., Tomilov, Y.: Optimal polynomial decay of functions and operator semigroups. Math. Ann. 347(2), 455–478 (2010)

    Article  MathSciNet  Google Scholar 

  2. Cao, X.-G., Liu, D.-Y., Xu, G.-Q.: Easy test for stability of laminated beams with structural damping and boundary feedback controls. J. Dyn. Control Syst. 13(3), 313–336 (2007)

    Article  MathSciNet  Google Scholar 

  3. Caputo, M.: Linear models of dissipation whose Q is almost frequency independent part II. Geophys. J. Int. 13, 529–539 (1967)

    Article  Google Scholar 

  4. Caputo, M.: Elasticity e Dissipazione (Elasticity and Dissipation). Zanichelli, Bologna (1969). (in Italian)

    Google Scholar 

  5. Caputo, M., Fabrizio, M.: 3D memory constitutive equations for plastic media. J. Eng. Mech. 143(5), D4016008 (2017)

    Article  Google Scholar 

  6. Caputo, M., Mainardi, F.: Linear models of dissipation in anelastic solids. Riv. Nuovo Cimento. (Ser. II) 1, 161–198 (1971)

    Article  Google Scholar 

  7. Choi, J., Maccamy, R.: Fractional order Volterra equations with applications to elasticity. J. Math. Anal. Appl. 139, 448–464 (1989)

    Article  MathSciNet  Google Scholar 

  8. Engel, K.J., Nagel, R.: One-parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics (With contributions by Brendle, S., Campiti, M., Hahn, T., Metafune, G., Nickel, G., Pallara, D., Perazzoli, C., Rhandi, A., Romanelli, S., Schnaubelt, R.). Springer, New York (2000)

    MATH  Google Scholar 

  9. Hansen, S.W.: In control and estimation of distributed parameter systems: non-linear phenomena. Int. Ser. Numer. Anal. 118, 143–170 (1994)

    Google Scholar 

  10. Hansen, S.W.: A Model for a Two-Layered Plate with Interfacial Slip. Birkhaüser, Basel (1994)

    Book  Google Scholar 

  11. Hansen, S.W., Spies, R.: Structural damping in a laminated beams due to interfacial slip. J. Sound Vib. 204, 183–202 (1997)

    Article  Google Scholar 

  12. Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204, p. xvi+523. Elsevier, Amsterdam (2006)

    Book  Google Scholar 

  13. Lo, A., Tatar, N.-E.: Stabilization of laminated beams with interfacial slip. Electron. J. Diff. Equ. 2015(129), 1–14 (2015)

    MathSciNet  MATH  Google Scholar 

  14. Maryati, T.K., Rivera, J.E.M., Rambaud, A., Vera, O.: Stability of an $N$-component Timoshenko beam with localized Kelvin–Voigt and frictional dissipation. Electron. J. Diff. Equ. 2018(136), 1–18 (2018)

    MathSciNet  MATH  Google Scholar 

  15. Mbodje, B.: Wave energy decay under fractional derivative controls. IMA J. Math. Control Inf. 23, 237–257 (2006)

    Article  MathSciNet  Google Scholar 

  16. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)

    Book  Google Scholar 

  17. Raposo, C.A.: Exponential stability for a structure with interfacial slip and frictional damping. Appl. Math. Lett. 53, 85–91 (2016)

    Article  MathSciNet  Google Scholar 

  18. Samko, S., Kilbas, A., Marichev, O.: Integral and Derivatives of Fractional Order. Gordon Breach, New York (1993)

    MATH  Google Scholar 

  19. Wang, J.-M., Xu, G.-Q., Yung, S.-P.: Exponential stabilization of laminated beams with structural damping and boundary feedback controls. SIAM J. Control Optim. 44(5), 1575–1597 (2005)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Mathematics Education Department, Islamic State University (UIN), South Tangerang, Indonesia

    Tita Maryati

  2. Departamento de Matemáticas, Universidad del Bío-Bío, Concepción, Chile

    Jaime Muñoz Rivera & Octavio Vera

  3. LNCC Petrópolis, Petrópolis, Brazil

    Jaime Muñoz Rivera

  4. Departamento de Matemáticas, Universidad de Chile, Santiago, Chile

    Verónica Poblete

Authors
  1. Tita Maryati
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  2. Jaime Muñoz Rivera
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  3. Verónica Poblete
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  4. Octavio Vera
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Correspondence to Verónica Poblete.

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The author is partially supported by CNPq-Brazil: Grants 158706/2014-5, 164793/2015-1 and 402689/2012-7. The author is partially supported by CNPq (PCI-LNCC/MCT); by project GI 171608/VC Universidad del Bío-Bío; and grateful to UIN, Indonesia, for its hospitality towards the visiting professors. The three previous authors are partially financed by Project Fondecyt 1191137.

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Maryati, T., Muñoz Rivera, J., Poblete, V. et al. Asymptotic Behavior in a Laminated Beams Due Interfacial Slip with a Boundary Dissipation of Fractional Derivative Type. Appl Math Optim 84, 85–102 (2021). https://doi.org/10.1007/s00245-019-09639-1

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  • DOI: https://doi.org/10.1007/s00245-019-09639-1

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