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doi: 10.3934/eect.2022001
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## Exponential stabilization of the problem of transmission of wave equation with linear dynamical feedback control

 School of Mathematical Sciences, Shanxi University, Taiyuan, 030006, China

*Corresponding author: Shugen Chai

Received  June 2021 Revised  October 2021 Early access January 2022

Fund Project: The first author is supported by the National Natural Science Foundation of China (No.11671240)

In this paper, we address exponential stabilization of transmission problem of the wave equation with linear dynamical feedback control. Using the classical energy method and multiplier technique, we prove that the energy of system exponentially decays.

Citation: Zhiling Guo, Shugen Chai. Exponential stabilization of the problem of transmission of wave equation with linear dynamical feedback control. Evolution Equations and Control Theory, doi: 10.3934/eect.2022001
##### References:
 [1] B. A. Akram and F. Mohamed, Stability result for viscoelastic wave equation with dynamic boundary conditions, Z. Angew. Math. Phys., 69 (2018), 95-95. [2] K. T. Andrews, K. L. Kuttler and M. Shillor, Second order evolution equations with dynamic boundary conditions, J. Math. Anal. Appl., 197 (1996), 781-795.  doi: 10.1006/jmaa.1996.0053. [3] H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, 1$^{st}$ edition, North-Holland Amsterdam, New York, 1973. [4] M. M. Cavalcanti, E. R. S. Coelho and V. N. D. Cavalcanti, Exponential stability for a transmission problem of a viscoelastic wave equation, Appl. Math. Optim., 81 (2020), 621-650.  doi: 10.1007/s00245-018-9514-9. [5] B. Chentouf and M. S. Boudellioua, A new approach to stabilization of the wave equation with boundary damping control, SQU Journal For Science, 9 (2004), 33-40. [6] F. Conrad and A. Mifdal, Uniform stabilization of a hybrid system with a class of nonlinear feedback laws, Adv. Math. Sci. Appl., 11 (2001), 549-569. [7] N. Fourrer and I. Lasiecka, Regularity and stability of a wave equation with a strong damping and dynamic boundary conditions, Evol. Equ. Control. The., 2 (2013), 631-667.  doi: 10.3934/eect.2013.2.631. [8] A. Haraux, Systèmes Dynamiques Dissipatifs et Applications, 1$^{st}$ edition, Masson, Paris, 1991. [9] M. A. Horn, Implications of sharp trace regularity results on boundary stabilization of the system of linear elasticity, J. Math. Anal. Appl., 223 (1998), 126-150.  doi: 10.1006/jmaa.1998.5963. [10] T. Kato, Perturbation Theory for Linear Operators, Reprint of the 1980 edition, Springer-Verlag, Berlin, 1995. [11] J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation, J. Differential Equations., 50 (1983), 163-182.  doi: 10.1016/0022-0396(83)90073-6. [12] I. Lasiecka and R. Triggiani, Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions, Appl. Math. Optim., 25 (1992), 189-224.  doi: 10.1007/BF01182480. [13] J. L. Lions, Contrólabilité Exacte Perturbations et Stabilization de Systèmes Distribués Tome 1 Contrólabilité Exacte, 1$^{st}$ edition, Masson, Paris, 1988. [14] W. J. Liu, Stabilization and controllability for the transmission wave equation, IEEE Trans. Automat. Control., 46 (2001), 1900-1907.  doi: 10.1109/9.975473. [15] W. J. Liu and G. H. Williams, The exponential stability of the problem of transmission of the wave equation, Bull. Austral. Math. Soc., 57 (1998), 305-327.  doi: 10.1017/S0004972700031683. [16] W. J. Liu and G. H. Williams, Exact controllability for problems of transmission of the plate equation with lower-order terms, Quart Appl. Math., 58 (2000), 37-68.  doi: 10.1090/qam/1738557. [17] W. J. Liu and G. H. Williams, Exact Neumann boundary controllability for problems of transmission of the wave equation, Glasg. Math. J., 41 (1999), 125-139.  doi: 10.1017/S0017089599970581. [18] Z. Q. Liu and Z. B. Fang, The global solvability and asymptotic behavior of a transmission problem for Kirchhoff-type wave equations with memory source on the boundary, Math. Methods Appl. Sci., 42 (2019), 6284-6300.  doi: 10.1002/mma.5722. [19] D. Mercier, S. Nicaise, M. A. Sammoury and A. Wehbe, Indirect stability of the wave equation with a dynamic boundary control, Math. Nachr., 291 (2018), 1114-1146.  doi: 10.1002/mana.201700021. [20] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, 1$^{st}$ edition, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [21] A. J. A. Ramos and M. W. P. Souza, Equivalence between observability at the boundary and stabilization for transmission problem of the wave equation, Z. Angew. Math. Phys., 68 (2017), Paper No. 48, 11 pp. doi: 10.1007/s00033-017-0791-y. [22] T. J. Xiao and J. Liang, Second order parabolic equations in Banach spaces with dynamic boundary conditions, Trans. Amer. Math. Soc., 356 (2004), 4787-4809.  doi: 10.1090/S0002-9947-04-03704-3. [23] Z. F. Zhang, Stabilization of the wave equation with variable coefficients and a dynamic boundary control, Electron. J. Differential Equations., 2016 (2016), Paper No. 27, 10 pp.

show all references

##### References:
 [1] B. A. Akram and F. Mohamed, Stability result for viscoelastic wave equation with dynamic boundary conditions, Z. Angew. Math. Phys., 69 (2018), 95-95. [2] K. T. Andrews, K. L. Kuttler and M. Shillor, Second order evolution equations with dynamic boundary conditions, J. Math. Anal. Appl., 197 (1996), 781-795.  doi: 10.1006/jmaa.1996.0053. [3] H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, 1$^{st}$ edition, North-Holland Amsterdam, New York, 1973. [4] M. M. Cavalcanti, E. R. S. Coelho and V. N. D. Cavalcanti, Exponential stability for a transmission problem of a viscoelastic wave equation, Appl. Math. Optim., 81 (2020), 621-650.  doi: 10.1007/s00245-018-9514-9. [5] B. Chentouf and M. S. Boudellioua, A new approach to stabilization of the wave equation with boundary damping control, SQU Journal For Science, 9 (2004), 33-40. [6] F. Conrad and A. Mifdal, Uniform stabilization of a hybrid system with a class of nonlinear feedback laws, Adv. Math. Sci. Appl., 11 (2001), 549-569. [7] N. Fourrer and I. Lasiecka, Regularity and stability of a wave equation with a strong damping and dynamic boundary conditions, Evol. Equ. Control. The., 2 (2013), 631-667.  doi: 10.3934/eect.2013.2.631. [8] A. Haraux, Systèmes Dynamiques Dissipatifs et Applications, 1$^{st}$ edition, Masson, Paris, 1991. [9] M. A. Horn, Implications of sharp trace regularity results on boundary stabilization of the system of linear elasticity, J. Math. Anal. Appl., 223 (1998), 126-150.  doi: 10.1006/jmaa.1998.5963. [10] T. Kato, Perturbation Theory for Linear Operators, Reprint of the 1980 edition, Springer-Verlag, Berlin, 1995. [11] J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation, J. Differential Equations., 50 (1983), 163-182.  doi: 10.1016/0022-0396(83)90073-6. [12] I. Lasiecka and R. Triggiani, Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions, Appl. Math. Optim., 25 (1992), 189-224.  doi: 10.1007/BF01182480. [13] J. L. Lions, Contrólabilité Exacte Perturbations et Stabilization de Systèmes Distribués Tome 1 Contrólabilité Exacte, 1$^{st}$ edition, Masson, Paris, 1988. [14] W. J. Liu, Stabilization and controllability for the transmission wave equation, IEEE Trans. Automat. Control., 46 (2001), 1900-1907.  doi: 10.1109/9.975473. [15] W. J. Liu and G. H. Williams, The exponential stability of the problem of transmission of the wave equation, Bull. Austral. Math. Soc., 57 (1998), 305-327.  doi: 10.1017/S0004972700031683. [16] W. J. Liu and G. H. Williams, Exact controllability for problems of transmission of the plate equation with lower-order terms, Quart Appl. Math., 58 (2000), 37-68.  doi: 10.1090/qam/1738557. [17] W. J. Liu and G. H. Williams, Exact Neumann boundary controllability for problems of transmission of the wave equation, Glasg. Math. J., 41 (1999), 125-139.  doi: 10.1017/S0017089599970581. [18] Z. Q. Liu and Z. B. Fang, The global solvability and asymptotic behavior of a transmission problem for Kirchhoff-type wave equations with memory source on the boundary, Math. Methods Appl. Sci., 42 (2019), 6284-6300.  doi: 10.1002/mma.5722. [19] D. Mercier, S. Nicaise, M. A. Sammoury and A. Wehbe, Indirect stability of the wave equation with a dynamic boundary control, Math. Nachr., 291 (2018), 1114-1146.  doi: 10.1002/mana.201700021. [20] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, 1$^{st}$ edition, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [21] A. J. A. Ramos and M. W. P. Souza, Equivalence between observability at the boundary and stabilization for transmission problem of the wave equation, Z. Angew. Math. Phys., 68 (2017), Paper No. 48, 11 pp. doi: 10.1007/s00033-017-0791-y. [22] T. J. Xiao and J. Liang, Second order parabolic equations in Banach spaces with dynamic boundary conditions, Trans. Amer. Math. Soc., 356 (2004), 4787-4809.  doi: 10.1090/S0002-9947-04-03704-3. [23] Z. F. Zhang, Stabilization of the wave equation with variable coefficients and a dynamic boundary control, Electron. J. Differential Equations., 2016 (2016), Paper No. 27, 10 pp.
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