# American Institute of Mathematical Sciences

June  2020, 9(2): 509-533. doi: 10.3934/eect.2020022

## Uniform stabilization of a wave equation with partial Dirichlet delayed control

 1 School of Mathematics, Tianjin University, Tianjin 300354, China 2 Department of Mathematics and Statistics, Qinghai Nationalities University, Xining, Qinghai 810007, China

* Corresponding author: Xiaorui Wang

Received  November 2018 Revised  July 2019 Published  June 2020 Early access  December 2019

Fund Project: This project is partially supported by the National Natural Science Foundation in China (NSFC 61773277), and partially supported by NSF of Qinghai Province (2017-ZJ-908)

In this paper, we consider the uniform stabilization of some high-dimensional wave equations with partial Dirichlet delayed control. Herein we design a parameterization feedback controller to stabilize the system. This is a new approach of controller design which overcomes the difficulty in stability analysis of the closed-loop system. The detailed procedure is as follows: At first we rewrite the system with partial Dirichlet delayed control into an equivalence cascaded system of a transport equation and a wave equation, and then we construct an exponentially stable target system; Further, we give the form of the parameterization feedback controller. To stabilize the system under consideration, we choose some appropriate kernel functions and define a bounded inverse linear transformation such that the closed-loop system is equivalent to the target system. Finally, we obtain the stability of closed-loop system by the stability of target system.

Citation: Xiaorui Wang, Genqi Xu. Uniform stabilization of a wave equation with partial Dirichlet delayed control. Evolution Equations and Control Theory, 2020, 9 (2) : 509-533. doi: 10.3934/eect.2020022
##### References:
 [1] K. Ammari, Dirichlet boundary stabilization of the wave equation, Asymptot. Anal., 30 (2002), 117-130. [2] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065.  doi: 10.1137/0330055. [3] D. Bresch-Pietri and M. Krstic, Adaptive compensation of diffusion-advection actuator dynamics using boundary measurements, Proc. of the 2015 Conference on Decision and Control, (2015). doi: 10.1109/CDC.2015.7402378. [4] G. Chen, Energy decay estimates and exact boundary valued controllability of the wave equation in a bounded domain, J. Math. Pure. Appl., 58 (1979), 249-273. [5] G. Chen, A note on boundary stabilization of the wave equation, SIAM J. Control Optim., 19 (1981), 106-113.  doi: 10.1137/0319008. [6] R. Datko, J. Lagnese and M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 24 (1986), 152-156.  doi: 10.1137/0324007. [7] R. Datko, Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM J. Control Optim., 26 (1988), 697-713.  doi: 10.1137/0326040. [8] R. Datko, Two examples of ill-posedness with respect to time delays revisited, IEEE Trans. Automat. Control, 42 (1997), 511-515.  doi: 10.1109/9.566660. [9] B. Z. Guo and X. Zhang, The regularity of the wave equation with partial Dirichlet control and collocated observation, SIAM J. Control Optim., 44 (2005), 1598-1613.  doi: 10.1137/040610702. [10] B. Z. Guo and Z. X. Zhang, Dirichlet boundary stabilization of the wave equation, ESAIM Control Optim. & Calc. Var., 13 (2007), 776-792.  doi: 10.1051/cocv:2007040. [11] W. Guo, H. C. Zhou and M. Krstic, Adaptive error feedback output regulation for 1D wave equation, Internat. J. Robust Nonlinear Control, 28 (2018), 4309-4329. [12] Z. J. Han and G. Q. Xu, Output-based stabilization of Euler-Bernoulli beam with time-delay in boundary input, IMA J. Math. Control Inform., 31 (2014), 533-550.  doi: 10.1093/imamci/dnt030. [13] I. Lasiecka, J. L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl., 65 (1986), 149-192. [14] I. Lasiecka and R. Triggiani, Dirichlet boundary stabilization of the wave equation with damping feedback of finite range, J. Math. Anal. Appl., 97 (1983), 112-130.  doi: 10.1016/0022-247X(83)90241-X. [15] I. Lasiecka and R. Triggiani, Uniform exponential energy decay of wave equations in a bounded region with $L_2 (0, +\infty; L_2(\Gamma))$-feedback control in the Dirichlet boundary conditions, J. Differential Equations, 66 (1987), 340-390.  doi: 10.1016/0022-0396(87)90025-8. [16] W. Littman, Boundary Control Theory for hyperbolic and parabolic partial differential equations with constant coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 5 (1978), 567-580. [17] X. F. Liu and G. Q. Xu, Exponential stabilization for Timoshenko beam with distributed delay in the boundary control, Abstr. Appl. Anal., (2013), Art. ID 726794, 15 pp. doi: 10.1155/2013/726794. [18] X. F. Liu and G. Q. Xu, Exponential stabilization for Timoshenko beam with different delays in the boundary control, IMA J. Math. Control Inform., 34 (2017), 93-110.  doi: 10.1093/imamci/dnv036. [19] C. A. McMillan, Stabilization of the wave equation with finite range Dirichlet boundary feedback, J. Math. Anal. Appl., 171 (1992), 139-155.  doi: 10.1016/0022-247X(92)90382-N. [20] S. Nicaise and C. Pingotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561-1585.  doi: 10.1137/060648891. [21] S. Nicaise and J. Valein, Stabilization of the wave equation on 1-d networks with a delay term in the nodal feedbacks, Netw. Heterog. Media, 2 (2007), 425-479.  doi: 10.3934/nhm.2007.2.425. [22] S. Nicaise and C. Pingotti, Exponential stability of second order evolution equations with structural damping and dynamic boundary delay feedback, IMA J. Math. Control Inform., 28 (2011), 417-446.  doi: 10.1093/imamci/dnr012. [23] S. Nicaise and C. Pignotti, Stabilization of second-order evolution equations with time delay, Math. Control Signals Systems, 26 (2014), 563-588.  doi: 10.1007/s00498-014-0130-1. [24] S. Nicaise and C. Pignotti, Exponential stability of abstract evolution equations with time delay, J. Evol. Equ., 15 (2015), 107-129.  doi: 10.1007/s00028-014-0251-5. [25] S. Nicaise and C. Pignotti, Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 791-813.  doi: 10.3934/dcdss.2016029. [26] Z.-H. Ning and Q.-X. Yan, Stabilization of the wave equation with variable coefficients and a delay in dissipative boundary feedback, J. Math. Anal. Appl., 367 (2010), 167-173.  doi: 10.1016/j.jmaa.2009.12.058. [27] C. Pignotti, A note on stabilization of locally damped wave equations with time delay, Systems Control Lett., 61 (2012), 92-97.  doi: 10.1016/j.sysconle.2011.09.016. [28] Y. F. Shang and G. Q. Xu, Stabilization of an Euler-Bernoulli beam with input delay in the boundary control, Systems Control Lett., 61 (2012), 1069-1078.  doi: 10.1016/j.sysconle.2012.07.012. [29] Y. F. Shang and G. Q. Xu, Dynamic feedback control and exponential stabilization of a compound system, J. Math. Anal. Appl., 422 (2015), 858-879.  doi: 10.1016/j.jmaa.2014.09.013. [30] A. Smyshlyaev, E. Cerpa and M. Krstic, Boundary stabilization of a 1-D wave equation with in-domain antidamping, SIAM J. Control Optim., 48 (2010), 4014-4031.  doi: 10.1137/080742646. [31] R. Triggiani, Exact boundary controllability on $L^2(\Omega)\times H^{-1}(\Omega)$ of the wave equation with Dirichlet boundary control acting on a portion of the boundary $\partial\Omega$, and Related Problems, Appl. Math. Optim., 18 (1988), 241-277.  doi: 10.1007/BF01443625. [32] H. Wang and G. Q. Xu, Exponential stabilization of 1-d wave equation with input delay, WSEAS Transactions on Mathematics, 12 (2013), 1001-1013. [33] G. Q. Xu, S. P. Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control, ESAIM Control Optim. Calc. Var., 12 (2006), 770-785.  doi: 10.1051/cocv:2006021. [34] G. Q. Xu and H. X. Wang, Stabilization of Timoshenko beam system with delay in the boundary control, Internat. J. Control, 86 (2013), 1165-1178.  doi: 10.1080/00207179.2013.787494. [35] P. F. Yao, On the observability inequalities for the exact controllability of the wave equation with variable coefficients, SIAM J. Control Optim., 37 (1999), 1568-1599.  doi: 10.1137/S0363012997331482. [36] P. F. Yao, Global smooth solutions for the quasilinear wave equation with boundary dissipation, J. Differential Equations, 241 (2007), 62-93.  doi: 10.1016/j.jde.2007.06.014. [37] P. F. Yao, Energy decay for the Cauchy problem of the linear wave equation of variable coefficients with dissipation, Chin. Ann. Math. Ser. B, 31 2010, 59–70. doi: 10.1007/s11401-008-0421-2. [38] P. F. Yao, Boundary controllability for the quasilinear wave equation, Appl. Math. Optim., 61 2010,191–233. doi: 10.1007/s00245-009-9088-7.

show all references

##### References:
 [1] K. Ammari, Dirichlet boundary stabilization of the wave equation, Asymptot. Anal., 30 (2002), 117-130. [2] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065.  doi: 10.1137/0330055. [3] D. Bresch-Pietri and M. Krstic, Adaptive compensation of diffusion-advection actuator dynamics using boundary measurements, Proc. of the 2015 Conference on Decision and Control, (2015). doi: 10.1109/CDC.2015.7402378. [4] G. Chen, Energy decay estimates and exact boundary valued controllability of the wave equation in a bounded domain, J. Math. Pure. Appl., 58 (1979), 249-273. [5] G. Chen, A note on boundary stabilization of the wave equation, SIAM J. Control Optim., 19 (1981), 106-113.  doi: 10.1137/0319008. [6] R. Datko, J. Lagnese and M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 24 (1986), 152-156.  doi: 10.1137/0324007. [7] R. Datko, Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM J. Control Optim., 26 (1988), 697-713.  doi: 10.1137/0326040. [8] R. Datko, Two examples of ill-posedness with respect to time delays revisited, IEEE Trans. Automat. Control, 42 (1997), 511-515.  doi: 10.1109/9.566660. [9] B. Z. Guo and X. Zhang, The regularity of the wave equation with partial Dirichlet control and collocated observation, SIAM J. Control Optim., 44 (2005), 1598-1613.  doi: 10.1137/040610702. [10] B. Z. Guo and Z. X. Zhang, Dirichlet boundary stabilization of the wave equation, ESAIM Control Optim. & Calc. Var., 13 (2007), 776-792.  doi: 10.1051/cocv:2007040. [11] W. Guo, H. C. Zhou and M. Krstic, Adaptive error feedback output regulation for 1D wave equation, Internat. J. Robust Nonlinear Control, 28 (2018), 4309-4329. [12] Z. J. Han and G. Q. Xu, Output-based stabilization of Euler-Bernoulli beam with time-delay in boundary input, IMA J. Math. Control Inform., 31 (2014), 533-550.  doi: 10.1093/imamci/dnt030. [13] I. Lasiecka, J. L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl., 65 (1986), 149-192. [14] I. Lasiecka and R. Triggiani, Dirichlet boundary stabilization of the wave equation with damping feedback of finite range, J. Math. Anal. Appl., 97 (1983), 112-130.  doi: 10.1016/0022-247X(83)90241-X. [15] I. Lasiecka and R. Triggiani, Uniform exponential energy decay of wave equations in a bounded region with $L_2 (0, +\infty; L_2(\Gamma))$-feedback control in the Dirichlet boundary conditions, J. Differential Equations, 66 (1987), 340-390.  doi: 10.1016/0022-0396(87)90025-8. [16] W. Littman, Boundary Control Theory for hyperbolic and parabolic partial differential equations with constant coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 5 (1978), 567-580. [17] X. F. Liu and G. Q. Xu, Exponential stabilization for Timoshenko beam with distributed delay in the boundary control, Abstr. Appl. Anal., (2013), Art. ID 726794, 15 pp. doi: 10.1155/2013/726794. [18] X. F. Liu and G. Q. Xu, Exponential stabilization for Timoshenko beam with different delays in the boundary control, IMA J. Math. Control Inform., 34 (2017), 93-110.  doi: 10.1093/imamci/dnv036. [19] C. A. McMillan, Stabilization of the wave equation with finite range Dirichlet boundary feedback, J. Math. Anal. Appl., 171 (1992), 139-155.  doi: 10.1016/0022-247X(92)90382-N. [20] S. Nicaise and C. Pingotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561-1585.  doi: 10.1137/060648891. [21] S. Nicaise and J. Valein, Stabilization of the wave equation on 1-d networks with a delay term in the nodal feedbacks, Netw. Heterog. Media, 2 (2007), 425-479.  doi: 10.3934/nhm.2007.2.425. [22] S. Nicaise and C. Pingotti, Exponential stability of second order evolution equations with structural damping and dynamic boundary delay feedback, IMA J. Math. Control Inform., 28 (2011), 417-446.  doi: 10.1093/imamci/dnr012. [23] S. Nicaise and C. Pignotti, Stabilization of second-order evolution equations with time delay, Math. Control Signals Systems, 26 (2014), 563-588.  doi: 10.1007/s00498-014-0130-1. [24] S. Nicaise and C. Pignotti, Exponential stability of abstract evolution equations with time delay, J. Evol. Equ., 15 (2015), 107-129.  doi: 10.1007/s00028-014-0251-5. [25] S. Nicaise and C. Pignotti, Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 791-813.  doi: 10.3934/dcdss.2016029. [26] Z.-H. Ning and Q.-X. Yan, Stabilization of the wave equation with variable coefficients and a delay in dissipative boundary feedback, J. Math. Anal. Appl., 367 (2010), 167-173.  doi: 10.1016/j.jmaa.2009.12.058. [27] C. Pignotti, A note on stabilization of locally damped wave equations with time delay, Systems Control Lett., 61 (2012), 92-97.  doi: 10.1016/j.sysconle.2011.09.016. [28] Y. F. Shang and G. Q. Xu, Stabilization of an Euler-Bernoulli beam with input delay in the boundary control, Systems Control Lett., 61 (2012), 1069-1078.  doi: 10.1016/j.sysconle.2012.07.012. [29] Y. F. Shang and G. Q. Xu, Dynamic feedback control and exponential stabilization of a compound system, J. Math. Anal. Appl., 422 (2015), 858-879.  doi: 10.1016/j.jmaa.2014.09.013. [30] A. Smyshlyaev, E. Cerpa and M. Krstic, Boundary stabilization of a 1-D wave equation with in-domain antidamping, SIAM J. Control Optim., 48 (2010), 4014-4031.  doi: 10.1137/080742646. [31] R. Triggiani, Exact boundary controllability on $L^2(\Omega)\times H^{-1}(\Omega)$ of the wave equation with Dirichlet boundary control acting on a portion of the boundary $\partial\Omega$, and Related Problems, Appl. Math. Optim., 18 (1988), 241-277.  doi: 10.1007/BF01443625. [32] H. Wang and G. Q. Xu, Exponential stabilization of 1-d wave equation with input delay, WSEAS Transactions on Mathematics, 12 (2013), 1001-1013. [33] G. Q. Xu, S. P. Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control, ESAIM Control Optim. Calc. Var., 12 (2006), 770-785.  doi: 10.1051/cocv:2006021. [34] G. Q. Xu and H. X. Wang, Stabilization of Timoshenko beam system with delay in the boundary control, Internat. J. Control, 86 (2013), 1165-1178.  doi: 10.1080/00207179.2013.787494. [35] P. F. Yao, On the observability inequalities for the exact controllability of the wave equation with variable coefficients, SIAM J. Control Optim., 37 (1999), 1568-1599.  doi: 10.1137/S0363012997331482. [36] P. F. Yao, Global smooth solutions for the quasilinear wave equation with boundary dissipation, J. Differential Equations, 241 (2007), 62-93.  doi: 10.1016/j.jde.2007.06.014. [37] P. F. Yao, Energy decay for the Cauchy problem of the linear wave equation of variable coefficients with dissipation, Chin. Ann. Math. Ser. B, 31 2010, 59–70. doi: 10.1007/s11401-008-0421-2. [38] P. F. Yao, Boundary controllability for the quasilinear wave equation, Appl. Math. Optim., 61 2010,191–233. doi: 10.1007/s00245-009-9088-7.
 [1] Yanni Guo, Genqi Xu, Yansha Guo. Stabilization of the wave equation with interior input delay and mixed Neumann-Dirichlet boundary. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2491-2507. doi: 10.3934/dcdsb.2016057 [2] Muhammad I. Mustafa. On the control of the wave equation by memory-type boundary condition. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 1179-1192. doi: 10.3934/dcds.2015.35.1179 [3] Boumediène Chentouf, Baowei Feng. On the stabilization of a flexible structure via a nonlinear delayed boundary control. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022032 [4] Kim Dang Phung. Boundary stabilization for the wave equation in a bounded cylindrical domain. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 1057-1093. doi: 10.3934/dcds.2008.20.1057 [5] Boumedièene Chentouf, Sabeur Mansouri. Boundary stabilization of a flexible structure with dynamic boundary conditions via one time-dependent delayed boundary control. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1127-1141. doi: 10.3934/dcdss.2021090 [6] Bopeng Rao, Laila Toufayli, Ali Wehbe. Stability and controllability of a wave equation with dynamical boundary control. Mathematical Control and Related Fields, 2015, 5 (2) : 305-320. doi: 10.3934/mcrf.2015.5.305 [7] Shitao Liu. Recovery of the sound speed and initial displacement for the wave equation by means of a single Dirichlet boundary measurement. Evolution Equations and Control Theory, 2013, 2 (2) : 355-364. doi: 10.3934/eect.2013.2.355 [8] Martin Gugat, Markus Dick. Time-delayed boundary feedback stabilization of the isothermal Euler equations with friction. Mathematical Control and Related Fields, 2011, 1 (4) : 469-491. doi: 10.3934/mcrf.2011.1.469 [9] Behzad Azmi, Karl Kunisch. Receding horizon control for the stabilization of the wave equation. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 449-484. doi: 10.3934/dcds.2018021 [10] Martin Gugat, Günter Leugering, Ke Wang. Neumann boundary feedback stabilization for a nonlinear wave equation: A strict $H^2$-lyapunov function. Mathematical Control and Related Fields, 2017, 7 (3) : 419-448. doi: 10.3934/mcrf.2017015 [11] Mohammad Akil, Ali Wehbe. Stabilization of multidimensional wave equation with locally boundary fractional dissipation law under geometric conditions. Mathematical Control and Related Fields, 2019, 9 (1) : 97-116. doi: 10.3934/mcrf.2019005 [12] Mokhtari Yacine. Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains. Evolution Equations and Control Theory, 2022, 11 (2) : 373-397. doi: 10.3934/eect.2021004 [13] Roberto Triggiani, Xiang Wan. From low to high-and lower-optimal regularity of the SMGTJ equation with Dirichlet and Neumann boundary control, and with point control, via explicit representation formulae. Evolution Equations and Control Theory, 2022  doi: 10.3934/eect.2022007 [14] Piotr Kowalski. The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2569-2580. doi: 10.3934/dcdsb.2014.19.2569 [15] Lassaad Aloui, Moez Khenissi. Boundary stabilization of the wave and Schrödinger equations in exterior domains. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 919-934. doi: 10.3934/dcds.2010.27.919 [16] Xu Zhang, Chuang Zheng, Enrique Zuazua. Time discrete wave equations: Boundary observability and control. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 571-604. doi: 10.3934/dcds.2009.23.571 [17] Khadijah Sharaf. A perturbation result for a critical elliptic equation with zero Dirichlet boundary condition. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1691-1706. doi: 10.3934/dcds.2017070 [18] Yaru Xie, Genqi Xu. Exponential stability of 1-d wave equation with the boundary time delay based on the interior control. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 557-579. doi: 10.3934/dcdss.2017028 [19] Xiaoqiang Dai, Chao Yang, Shaobin Huang, Tao Yu, Yuanran Zhu. Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems. Electronic Research Archive, 2020, 28 (1) : 91-102. doi: 10.3934/era.2020006 [20] Zhiling Guo, Shugen Chai. Exponential stabilization of the problem of transmission of wave equation with linear dynamical feedback control. Evolution Equations and Control Theory, 2022  doi: 10.3934/eect.2022001

2020 Impact Factor: 1.081