Well-posedness and stability results in a Timoshenko-type system of thermoelasticity of type III with delay
Tóm tắt
In this paper, we consider a one-dimensional linear thermoelastic system of Timoshenko type with delay, where the heat conduction is given by Green and Naghdi’s theory. We establish the well-posedness and the stability of the system for the cases of equal and nonequal speeds of wave propagation.
Tài liệu tham khảo
Timoshenko S.: On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philos. Mag. 41, 744–746 (1921)
Kim J.U., Renardy Y.: Boundary control of the Timoshenko beam. SIAM J. Control Optim. 25(6), 1417–1429 (1987)
Raposo C.A., Ferreira J., Santos M.L., Castro N.N.O.: Exponential stability for the Timoshenko system with two weak dampings. Appl. Math. Lett. 18, 535–541 (2005)
Soufyane A., Wehbe A.: Uniform stabilization for the Timoshenko beam by a locally distributed damping. Electron. J. Differ. Equ. 2003(29), 1–14 (2003)
Muñoz Rivera J.E., Racke R.: Global stability for damped Timoshenko systems. Discret. Contin. Dyn. Syst. 9(6), 1625–1639 (2003)
Muñoz Rivera J.E., Racke R.: Timoshenko systems with indefinite damping. J. Math. Anal. Appl. 341, 1068–1083 (2008)
Mustafa M.I., Messaoudi S.A.: General energy decay rates for a weakly damped Timoshenko system. J. Dyn. Control Syst. 16, 211–226 (2010)
Ammar-Khodja F., Benabdallah A., Muñoz Rivera J.E., Racke R.: Energy decay for Timoshenko systems of memory type. J. Differ. Equ. 194(1), 82–115 (2003)
Guesmia A., Messaoudi S.A.: General energy decay estimates of Timoshenko systems with frictional versus viscoelastic damping. Math. Methods Appl. Sci. 32, 2102–2122 (2009)
Fernández Sare H.D., Muñnoz Rivera J.E.: Stability of Timoshenko systems with past history. J. Math. Anal. Appl. 339 # 1, 482–502 (2008)
Muñoz Rivera J.E., Racke R.: Mildly dissipative nonlinear Timoshenko systems-global existence and exponential stability. J. Math. Anal. Appl. 276, 248–276 (2002)
Green A.E., Naghdi P.M.: A re-examination of the basic postulates of thermomechanics. Proc. R. Soc. Lond. A. 432, 171–194 (1991)
Green A.E., Naghdi P.M.: On undamped heat waves in an elastic solid. J. Therm. Stress. 15, 253–264 (1992)
Green A.E., Naghdi P.M.: Thermoelasticity without energy dissipation. J. Elast. 31, 189–208 (1993)
Chandrasekharaiah D.S.: Hyperbolic thermoelasticity: a review of recent literature. Appl. Mech. Rev. 51, 705–729 (1998)
Messaoudi S.A., Said-Houari B.: Energy decay in a Timoshenko-type system of thermoelasticity of type III. JMAA 348, 298–307 (2008)
Messaoudi S.A., Fareh A.: Energy decay in a Timoshenko-type system of thermoelasticity of type III with different wave-propagation speeds. Arab. J. Math. 2, 199–207 (2013)
Kafini M.: General energy decay in a Timoshenko-type system of thermoelasticity of type III with a viscoelastic damping. J. Math. Anal. Appl. 375, 523–537 (2011)
Suh I.H., Bien Z.: Use of time delay action in the controller design. IEEE Trans. Automat. Control. 25, 600–603 (1980)
Datko R.: Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks. SIAM J. Control Optim. 26(3), 697–713 (1988)
Zuazua E.: Exponential decay for the semi-linear wave equation with locally distributed damping. Comm. Partial Differ. Equ. 15, 205–235 (1990)
Nicaise S., Pignotti C.: Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM J. Control Optim. 45(5), 1561–1585 (2006)
Ait Benhassi E.M., Ammari K., Boulite S., Maniar L.: Feedback stabilization of a class of evolution equations with delay. J. Evol. Equ. 9, 103–121 (2009)
Nicaise S., Pignotti C., Valein J.: Exponential stability of the wave equation with boundary time-varying delay. Discret. Contin. Dyn. Syst. 2(3), 559–581 (2009)
Racke R.: Instability of coupled systems with delay. Commun. Pure Appl. Anal. 11(5), 1753–1773 (2012)
Guesmia A., Messaoudi S., Soufyane A.: Stabilization of a linear Timoshenko system with infinite history and applications to the Timoshenko-heat systems. Electron. J. Differ. Equ. 193, 1–45 (2012)
Guesmia A.: Well-posedness and exponential stability of an abstract evolution equation with infinite memory and time delay. IMA J. Math. Control Inf. 30, 507–526 (2013)
Said-Houari B., Soufyane A.: Stability result of the Timoshenko system with delay and boundary feedback. IMA J. Math. Control Inform. 29(3), 383–398 (2012)
Kirane M., Said-Houari B., Anwar M.: Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks. Commun. Pure Appl. Anal. 10(2), 667–686 (2011)
Said-Houari B., Laskri Y.: A stability result of a Timoshenko system with a delay term in the internal feedback. Appl. Math. Comput. 217(6), 2857–2869 (2010)
Brezis H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, Berlin (2010)
