American Institute of Mathematical Sciences

Advanced
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 & 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.   Google Scholar

[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.  Google Scholar

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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.   Google Scholar

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G. Chen, A note on boundary stabilization of the wave equation, SIAM J. Control Optim., 19 (1981), 106-113.  doi: 10.1137/0319008.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

K. Ammari, Dirichlet boundary stabilization of the wave equation, Asymptot. Anal., 30 (2002), 117-130.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[5]

G. Chen, A note on boundary stabilization of the wave equation, SIAM J. Control Optim., 19 (1981), 106-113.  doi: 10.1137/0319008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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