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doi: 10.3934/dcdsb.2021053
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## Dynamics of Timoshenko system with time-varying weight and time-varying delay

 1 Department of Mathematics, Federal University of Bahia, Salvador, 40170-115, Bahia, Brazil 2 Faculty of Exact Sciences and Technology, Federal University of Pará, Manoel de Abreu Street, s/n, 68440-000, Abaetetuba, Pará, Brazil 3 Department of Mathematics, Federal University of São João del-Rei, São João del-Rei, 36307-352, Minas Gerais, Brazil

* Corresponding author: raposo@ufsj.edu.br

Received  May 2020 Revised  November 2020 Early access February 2021

Fund Project: The first author was partially supported by CAPES (Brazil)

This paper is concerned with the well-posedness of global solution and exponential stability to the Timoshenko system subject with time-varying weights and time-varying delay. We consider two problems: full and partially damped systems. We prove existence of global solution for both problems combining semigroup theory with the Kato's variable norm technique. To prove exponential stability, we apply the Energy Method. For partially damped system the exponential stability is proved under assumption of equal-speed wave propagation in the transversal and angular directions. For full damped system the exponential stability is obtained without the hypothesis of equal-speed wave propagation.

Citation: Carlos Nonato, Manoel Jeremias dos Santos, Carlos Raposo. Dynamics of Timoshenko system with time-varying weight and time-varying delay. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021053
##### References:
 [1] F. Ali Mehmeti, Nonlinear Waves in Networks, vol 80, Mathematical Research, Akademie-Verlag, Berlim, 1994.  Google Scholar [2] F. Ammar-Khodja, A. Benabdallah, J. E. Muñnoz Rivera and R. Racke, Energy decay for Timoshenko systems of memory type, Journal of Differential Equations, 194 (2003), 82-115.  doi: 10.1016/S0022-0396(03)00185-2.  Google Scholar [3] V. Barros, C. Nonato and C. Raposo, Global existence and energy decay of solutions for a wave equation with non-constant delay and nonlinear weights, Electronic Research Archive, 28 (2020), 205-220.  doi: 10.3934/era.2020014.  Google Scholar [4] R. Datko, Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM Journal on Control and Optimization, 26 (1988), 697-713.  doi: 10.1137/0326040.  Google Scholar [5] R. Datko, J. Lagnese and M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM Journal on Control and Optimization, 24 (1986), 152-156.  doi: 10.1137/0324007.  Google Scholar [6] B. Feng and M. L. Pelicer, Global existence and exponential stability for a nonlinear Timoshenko system with delay, Boundary Value Problems, 24 (1986).  doi: 10.1186/s13661-015-0468-4.  Google Scholar [7] A. Guesmia, Well-posedness and exponential stability of an abstract evolution equation with infinity memory and time delay, IMA Journal of Mathematical Control and Information, 30 (2013), 507-526.  doi: 10.1093/imamci/dns039.  Google Scholar [8] A. Haraux, Two remarks on dissipative hyperbolic problems, Research Notes in Mathematics Pitman, Boston, MA, 122 1985,161–179.  Google Scholar [9] T. Kato, Linear and Quasilinear Equations of Evolution of Hyperbolic Type, C.I.M.E. Summer Sch., 72, Springer, Heidelberg, 2011,125-191. doi: 10.1007/978-3-642-11105-1_4.  Google Scholar [10] T. Kato, Abstract Differential Equations and Nonlinear Mixed Problems, Lezioni Fermiane, Scuola Normale Superiore, Pisa, 1985.  Google Scholar [11] M. Kirane, B. Said-Houari and M. N. Anwar, Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks, Communications on Pure and Applied Analysis, 10 (2011), 667-686.  doi: 10.3934/cpaa.2011.10.667.  Google Scholar [12] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson-John Wiley, Paris, 1994.  Google Scholar [13] F. Z. Mahdi and A. Hakem, Global existence and asymptotic stability for the initial boundary value problem of the linear Bresse system with a time-varying delay term, Journal of Partial Differential Equations, 32 (2019), 93-111.  doi: 10.4208/jpde.v32.n2.1.  Google Scholar [14] S. Nicaise and C. Pignotti, Interior feedback stabilization of wave equations with time dependence delay, Electronic Journal of Differential Equations, 41 (2011), 1-20.   Google Scholar [15] S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay, Differential Integral Equations, 21 (2008), 935-958.   Google Scholar [16] S. Nicaise, C. Pignotti and J. Valein, Exponential stability of the wave equation with boundary time-varying delay, Discrete and Continuous Dynamical Systems Series S, 4 (2011), 693-722.  doi: 10.3934/dcdss.2011.4.693.  Google Scholar [17] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Vol. 44 of Applied Mathematics Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar [18] J. E. M. Rivera and R. Racke, Global stability for damped Timoshenko systems, Discrete Continuous and Dynamical Systems, 9 (2003), 1625-1639.  doi: 10.3934/dcds.2003.9.1625.  Google Scholar [19] B. Said-Houari and Y. Laskri, A stability result of a Timoshenko system with a delay term in the internal feedback, Applied Mathematics and Computation, 217 (2010), 2857-2869.  doi: 10.1016/j.amc.2010.08.021.  Google Scholar [20] A. Soufyane, Stabilisation de la poutre de Timoshenko, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 328 (1999), 731-734.  doi: 10.1016/S0764-4442(99)80244-4.  Google Scholar [21] N. G. Stephen, The second frequency spectrum of Timoshenko beams theory - Further assessment, Journal of Sound and Vibration, 292 (2006), 372-389.  doi: 10.1016/j.jsv.2005.08.003.  Google Scholar [22] N. G. Stephen and S. Puchegger, On the valid frequency range of Timoshenko beam theory, Journal of Sound and Vibration, 3 (2006), 1082-1087.  doi: 10.1016/j.jsv.2006.04.020.  Google Scholar [23] G. Q. Xu, S. P Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control, ESAIM Control Optimisation and Calculus of Variations, 12 (2006), 770-785.  doi: 10.1051/cocv:2006021.  Google Scholar [24] X-G Yang, J. Zhang and Y. Lu, Dynamics of the nonlinear Timoshenko system with variable delay, Applied Mathematics and Optimization, 2018 (2018).  doi: 10.1007/s00245-018-9539-0.  Google Scholar

show all references

##### References:
 [1] F. Ali Mehmeti, Nonlinear Waves in Networks, vol 80, Mathematical Research, Akademie-Verlag, Berlim, 1994.  Google Scholar [2] F. Ammar-Khodja, A. Benabdallah, J. E. Muñnoz Rivera and R. Racke, Energy decay for Timoshenko systems of memory type, Journal of Differential Equations, 194 (2003), 82-115.  doi: 10.1016/S0022-0396(03)00185-2.  Google Scholar [3] V. Barros, C. Nonato and C. Raposo, Global existence and energy decay of solutions for a wave equation with non-constant delay and nonlinear weights, Electronic Research Archive, 28 (2020), 205-220.  doi: 10.3934/era.2020014.  Google Scholar [4] R. Datko, Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM Journal on Control and Optimization, 26 (1988), 697-713.  doi: 10.1137/0326040.  Google Scholar [5] R. Datko, J. Lagnese and M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM Journal on Control and Optimization, 24 (1986), 152-156.  doi: 10.1137/0324007.  Google Scholar [6] B. Feng and M. L. Pelicer, Global existence and exponential stability for a nonlinear Timoshenko system with delay, Boundary Value Problems, 24 (1986).  doi: 10.1186/s13661-015-0468-4.  Google Scholar [7] A. Guesmia, Well-posedness and exponential stability of an abstract evolution equation with infinity memory and time delay, IMA Journal of Mathematical Control and Information, 30 (2013), 507-526.  doi: 10.1093/imamci/dns039.  Google Scholar [8] A. Haraux, Two remarks on dissipative hyperbolic problems, Research Notes in Mathematics Pitman, Boston, MA, 122 1985,161–179.  Google Scholar [9] T. Kato, Linear and Quasilinear Equations of Evolution of Hyperbolic Type, C.I.M.E. Summer Sch., 72, Springer, Heidelberg, 2011,125-191. doi: 10.1007/978-3-642-11105-1_4.  Google Scholar [10] T. Kato, Abstract Differential Equations and Nonlinear Mixed Problems, Lezioni Fermiane, Scuola Normale Superiore, Pisa, 1985.  Google Scholar [11] M. Kirane, B. Said-Houari and M. N. Anwar, Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks, Communications on Pure and Applied Analysis, 10 (2011), 667-686.  doi: 10.3934/cpaa.2011.10.667.  Google Scholar [12] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson-John Wiley, Paris, 1994.  Google Scholar [13] F. Z. Mahdi and A. Hakem, Global existence and asymptotic stability for the initial boundary value problem of the linear Bresse system with a time-varying delay term, Journal of Partial Differential Equations, 32 (2019), 93-111.  doi: 10.4208/jpde.v32.n2.1.  Google Scholar [14] S. Nicaise and C. Pignotti, Interior feedback stabilization of wave equations with time dependence delay, Electronic Journal of Differential Equations, 41 (2011), 1-20.   Google Scholar [15] S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay, Differential Integral Equations, 21 (2008), 935-958.   Google Scholar [16] S. Nicaise, C. Pignotti and J. Valein, Exponential stability of the wave equation with boundary time-varying delay, Discrete and Continuous Dynamical Systems Series S, 4 (2011), 693-722.  doi: 10.3934/dcdss.2011.4.693.  Google Scholar [17] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Vol. 44 of Applied Mathematics Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar [18] J. E. M. Rivera and R. Racke, Global stability for damped Timoshenko systems, Discrete Continuous and Dynamical Systems, 9 (2003), 1625-1639.  doi: 10.3934/dcds.2003.9.1625.  Google Scholar [19] B. Said-Houari and Y. Laskri, A stability result of a Timoshenko system with a delay term in the internal feedback, Applied Mathematics and Computation, 217 (2010), 2857-2869.  doi: 10.1016/j.amc.2010.08.021.  Google Scholar [20] A. Soufyane, Stabilisation de la poutre de Timoshenko, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 328 (1999), 731-734.  doi: 10.1016/S0764-4442(99)80244-4.  Google Scholar [21] N. G. Stephen, The second frequency spectrum of Timoshenko beams theory - Further assessment, Journal of Sound and Vibration, 292 (2006), 372-389.  doi: 10.1016/j.jsv.2005.08.003.  Google Scholar [22] N. G. Stephen and S. Puchegger, On the valid frequency range of Timoshenko beam theory, Journal of Sound and Vibration, 3 (2006), 1082-1087.  doi: 10.1016/j.jsv.2006.04.020.  Google Scholar [23] G. Q. Xu, S. P Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control, ESAIM Control Optimisation and Calculus of Variations, 12 (2006), 770-785.  doi: 10.1051/cocv:2006021.  Google Scholar [24] X-G Yang, J. Zhang and Y. Lu, Dynamics of the nonlinear Timoshenko system with variable delay, Applied Mathematics and Optimization, 2018 (2018).  doi: 10.1007/s00245-018-9539-0.  Google Scholar
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