Global existence and energy decay of solutions to a viscoelastic Timoshenko beam system with a nonlinear delay term

References

• Timoshenko S. On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philos. Mag. 1921;41:744–746.
   Google Scholar
• Datko R, Lagnese J, Polis MP. An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM J. Control Optim. 1986;24:152–156.
   Web of Science ®Google Scholar
• Cavalcanti MM, Cavalcanti VD, Lasiecka I. Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction. J. Diff. Equ. 2007;236:407–459.
   Web of Science ®Google Scholar
• Park JY, Park SH. General decay for a nonlinear beam equation with weak dissipation. J. Math. Phys. 2010;51:1–8. 073508.
   Web of Science ®Google Scholar
• Kim JU, Renardy Y. Boundary control of the Timoshenko beam. SIAM J. Control Optim. 1987;25:1417–1429.
   Web of Science ®Google Scholar
• Messaoudi SA, Mustafa MI. On the stabilization of the Timoshenko system by a weak nonlinear dissipation. Math. Meth. Appl. Sci. 2009;32:454–469.
   Web of Science ®Google Scholar
• Muñoz Rivera JE, Racke R. Mildly dissipative nonlinear Timoshenko systems:global existence and exponential stability. J. Math. Anal. Appl. 2002;276:248–276.
   Web of Science ®Google Scholar
• Muñoz Rivera JE, Racke R. Global stability for damped Timoshenko systems. Discrete Contin. Dyn. Syst. 2003;9:1625–1639.
   Web of Science ®Google Scholar
• Alabau-Boussouira F. Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control. Nonlinear Differ. Equ. Appl. 2007;14:643–669.
   Web of Science ®Google Scholar
• Abdallah C, Dorato P, Benitez-Read J, et al. Delayed positive feedback can stabilize oscillatory system. San Francisco: ACC; 1993. p. 3106–3107.
   Google Scholar
• Suh IH, Bien Z. Use of time delay action in the controller design. IEEE Trans. Autom. Control. 1980;25:600–603.
   Web of Science ®Google Scholar
• Benaissa A, Bahlil M. Global existence and energy decay of solutions to a nonlinear Timoshenko beam system with a delay term. Taiwanese J. Math. 2014;18:1411–1437.
   Web of Science ®Google Scholar
• 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. 2006;45:1561–1585.
   Web of Science ®Google Scholar
• Laskri Y, Said-Houari B. A stability result of a Timoshenko system with a delay term in the internal feedback. Appl. Math. Comput. 2010;217:2857–2869.
   Web of Science ®Google Scholar
• Benaissa A, Benguessoum A, Messaoudi SA. Global existence and energy decay of solutions to a viscoelastic wave equation with a delay term in the non-linear internal feedback. Int. J. Dyn. Syst. Differ. Equ. 2014;5:1–26.
   Google Scholar
• Guesmia A, Messaoudi SA. General energy decay estimates of Timoshenko systems with frictional versus viscoelastic damping. Math. Meth. Appl. Sci. 2009;32:2102–2122.
   Web of Science ®Google Scholar
• Lions JL. Quelques méthodes de résolution des problèmes aux limites non linéaires. Paris: Dunod; 1969.
   Google Scholar
• Lasiecka I, Tataru D. Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping. Differ. Integral Equ. 1993;6:507–533.
   Google Scholar
• Liu WJ, Zuazua E. Decay rates for dissipative wave equations. Papers in memory of Ennio De Giorgi (Italian). Ricerche Mat. 1999;48:61–75.
   Google Scholar
• Alabau-Boussouira F. On convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems. Appl. Math. Optim. 2005;51:61–105.
   Web of Science ®Google Scholar
• Chen G. Control and stabilization for the wave equation in a bounded domain, Part I. SIAM J. Control Optim. 1979;17:66–81.
   Web of Science ®Google Scholar
• Chen G. Control and stabilization for the wave equation in a bounded domain, Part II. SIAM J. Control Optim. 1981;19:114–122.
   Web of Science ®Google Scholar
• Eller M, Lagnese JE, Nicaise S. Decay rates for solutions of a Maxwell system with nonlinear boundary damping. Comput. Appl. Math. 2002;21:135–165.
   Web of Science ®Google Scholar
• Han ZJ, Xu GQ. Exponential stability of Timoshenko beam system with delay terms in boundary feedbacks. ESAIM Control Optim. 2011;17:552–574.
   Web of Science ®Google Scholar
• Haraux A. Two remarks on dissipative hyperbolic problems. Vol. 122, Research notes in mathematics. Boston (MA): Pitman; 1985. p. 161–179.
   Google Scholar
• Kato T. Linear and quasilinear equations of evolution of hyperbolic type. In: Hyperbolicity, C.I.M.E. Summer Sch., Vol. 72, Heidelberg: Springer; 2011. p. 125–191.
   Google Scholar
• Komornik V. Exact controllability and stabilization.The multiplier method. Paris: Masson--John Wiley; 1994.
   Google Scholar
• Arnold VI. Mathematical methods of classical mechanics. New York (NY): Springer-Verlag; 1989.
   Google Scholar
• Benaissa A, Guesmia A. Energy decay for wave equations of φ-Laplacian type with weakly nonlinear dissipation. Electron. J. Differ. Equ. 2008;2008:1–22.
   Google Scholar
• Zhong QC. Robust control of time-delay systems. London: Springer; 2006.
   Google Scholar