Uniform stabilization of a wave equation with partial Dirichlet delayed control

Previous Article
Null controllability for a heat equation with dynamic boundary conditions and drift terms
EECT Home
This Issue
Next Article
On a Kirchhoff wave model with nonlocal nonlinear damping

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

Uniform stabilization of a wave equation with partial Dirichlet delayed control

Xiaorui Wang ^1,2,, and Genqi Xu ^1,

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.

Keywords: Wave equation, delayed control, Dirichlet type boundary, boundary stabilization.

Mathematics Subject Classification: Primary: 93C20, 93D15; Secondary: 35G15, 35L05.

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

Tools

Metrics

Other articles
by authors

on AIMS
Xiaorui Wang Genqi Xu
on Google Scholar
Xiaorui Wang Genqi Xu

2020 Impact Factor: 1.081

Tools

Article outline

2020 Impact Factor: 1.081

Tools

Metrics

Other articles
by authors

on AIMS
Xiaorui Wang Genqi Xu
on Google Scholar
Xiaorui Wang Genqi Xu

[Back to Top]