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On a Kirchhoff wave model with nonlocal nonlinear damping
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 |
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.
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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. |
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