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Exponential stability of 1-d wave equation with the boundary time delay based on the interior control
Department of Mathematics, Tianjin University, Haihe Education Park, Tianjin, Tianjin 300350, China |
In this paper, the stability problem of 1-d wave equation with the boundary delay and the interior control is considered. The well-posedness of the closed-loop system is investigated by the linear operator. Based on the idea of Lyapunov functional technology, we give the condition on the relationship between the control parameter α and the delay parameter k to guarantee the exponential stability of the system.
References:
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K. Ammari, S. Nicaise and C. Pignotti,
Feedback boundary stabilization of wave equations with interior delay, Systems Control Letters, 59 (2010), 623-628.
doi: 10.1016/j.sysconle.2010.07.007. |
| [2] |
G. Abdallah, P. Dorato, J. Benitez and R. Byrne, Delayed positive feedback can stabilize oscillatory systems, in ACC (American control conference), San Francisco, (1993), 3106– 3107. |
| [3] |
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. |
| [4] |
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. |
| [5] |
R. Datko,
Two examples of ill-posedness with respect to time delays revisited, IEEE Trans. Autom Control, 42 (1997), 511-515.
doi: 10.1109/9.566660. |
| [6] |
R. Datko,
Two questions concerning the boundary control of certain elastic systems, Journal of Differential Equations, 92 (1991), 27-44.
doi: 10.1016/0022-0396(91)90062-E. |
| [7] |
R. Datko, J. Lagnese and M. P. Polis,
An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM Journal on Control and Optimization, 24 (1986), 152-156.
doi: 10.1137/0324007. |
| [8] |
S. Gerbi and B. Said-Houari,
Existence and exponential stability of a damped wave equation with dynamic boundary conditions and a delay term, Applied Mathematics Computation, 218 (2012), 11900-11910.
doi: 10.1016/j.amc.2012.05.055. |
| [9] |
Y. N. Guo and G. Q. Xu, Stabilization of wave equations with boundary delays and non-collocated feedback controls, 24th Chinese Control and Decision Conference, 2012. |
| [10] |
W. H. Kwon, G. W. Lee and S. W. Kim,
Performance improvement, using time delays in multi-variable controller design, INT J. Control, 52 (1990), 1455-1473.
|
| [11] |
J. Li and S. Chai,
Energy decay for a nonlinear wave equation of variable coefficients with acoustic boundary conditions and a time-varying delay in the boundary feedback, Nonlinear Analysis Theory Methods Applications, 112 (2015), 105-117.
doi: 10.1016/j.na.2014.08.021. |
| [12] |
W. Liu, General decay of the solution for a viscoelastic wave equation with a time-varying delay term in the internal feedback, Journal of Mathematical Physics, 54 (2012), 043504, 9 pp.
doi: 10.1063/1.4799929. |
| [13] |
J. Lutzen,
Euler's vision of a general partial differential calculus for a generalized kind of function, Mathematics Magazine, 56 (1983), 299-306.
doi: 10.2307/2690370. |
| [14] |
O. Morgul,
On the stabilization and stability robustness against small delays of some damped wave equations, Automatic Control IEEE Transactions on, 40 (1995), 1626-1630.
doi: 10.1109/9.412634. |
| [15] |
W. L. Miranker,
The wave equation in a medium in motion, IBM Journal of Research and Development, 4 (1960), 36-42.
doi: 10.1147/rd.41.0036. |
| [16] |
S. Nicaise and C. Pignotti,
Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, Siam Journal on Control Optimization, 45 (2006), 1561-1585.
doi: 10.1137/060648891. |
| [17] |
S. Nicaise,
Interior feedback stabilization of wave equations with time dependent delay, Electronic Journal of Differential Equations, 41 (2011), 1-20.
doi: 10.1016/j.sysconle.2011.09.016. |
| [18] |
S. Nicaise and J. Valein,
Stabilization of the wave equation on 1-d networks with a delay term in the nodal feedbacks, Networks and Heterogeneous Midea, 2 (2007), 425-479.
doi: 10.3934/nhm.2007.2.425. |
| [19] |
S. Nicaise and J. Valein,
Stability of the heat and of the wave equations with boundary time-varying delays, Discrete and Continuous Dynamical Systems-Series S, 2 (2009), 559-581.
doi: 10.3934/dcdss.2009.2.559. |
| [20] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. |
| [21] |
C. Pignotti,
A note on stabilization of locally damped wave equations with time delay, Systems Control Letters, 61 (2012), 92-97.
doi: 10.1016/j.sysconle.2011.09.016. |
| [22] |
I. Suh and Z. Bien,
Use of time-delay actions in the controller design, IEEE Transactions on Automatic Control, 25 (1980), 600-603.
|
| [23] |
Y. Shang and G. Q. Xu,
The stability of a wave equation with delay-dependent position, IMA Journal of Mathematical Control and Information, 28 (2011), 75-95.
doi: 10.1093/imamci/dnq026. |
| [24] |
H. Wang and G. Xu,
Exponential stabilization of 1-d wave equation with input delay, Wseas Transactions on Mathematics, 12 (2013), 1001-1013.
|
| [25] |
G. Q. Xu, S. Yung and L. Li,
Stabilization of wave systems with input delay in the boundary control, Control, Optimisation and Calculus of Variations, 12 (2006), 770-785.
doi: 10.1051/cocv:2006021. |
show all references
References:
| [1] |
K. Ammari, S. Nicaise and C. Pignotti,
Feedback boundary stabilization of wave equations with interior delay, Systems Control Letters, 59 (2010), 623-628.
doi: 10.1016/j.sysconle.2010.07.007. |
| [2] |
G. Abdallah, P. Dorato, J. Benitez and R. Byrne, Delayed positive feedback can stabilize oscillatory systems, in ACC (American control conference), San Francisco, (1993), 3106– 3107. |
| [3] |
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. |
| [4] |
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. |
| [5] |
R. Datko,
Two examples of ill-posedness with respect to time delays revisited, IEEE Trans. Autom Control, 42 (1997), 511-515.
doi: 10.1109/9.566660. |
| [6] |
R. Datko,
Two questions concerning the boundary control of certain elastic systems, Journal of Differential Equations, 92 (1991), 27-44.
doi: 10.1016/0022-0396(91)90062-E. |
| [7] |
R. Datko, J. Lagnese and M. P. Polis,
An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM Journal on Control and Optimization, 24 (1986), 152-156.
doi: 10.1137/0324007. |
| [8] |
S. Gerbi and B. Said-Houari,
Existence and exponential stability of a damped wave equation with dynamic boundary conditions and a delay term, Applied Mathematics Computation, 218 (2012), 11900-11910.
doi: 10.1016/j.amc.2012.05.055. |
| [9] |
Y. N. Guo and G. Q. Xu, Stabilization of wave equations with boundary delays and non-collocated feedback controls, 24th Chinese Control and Decision Conference, 2012. |
| [10] |
W. H. Kwon, G. W. Lee and S. W. Kim,
Performance improvement, using time delays in multi-variable controller design, INT J. Control, 52 (1990), 1455-1473.
|
| [11] |
J. Li and S. Chai,
Energy decay for a nonlinear wave equation of variable coefficients with acoustic boundary conditions and a time-varying delay in the boundary feedback, Nonlinear Analysis Theory Methods Applications, 112 (2015), 105-117.
doi: 10.1016/j.na.2014.08.021. |
| [12] |
W. Liu, General decay of the solution for a viscoelastic wave equation with a time-varying delay term in the internal feedback, Journal of Mathematical Physics, 54 (2012), 043504, 9 pp.
doi: 10.1063/1.4799929. |
| [13] |
J. Lutzen,
Euler's vision of a general partial differential calculus for a generalized kind of function, Mathematics Magazine, 56 (1983), 299-306.
doi: 10.2307/2690370. |
| [14] |
O. Morgul,
On the stabilization and stability robustness against small delays of some damped wave equations, Automatic Control IEEE Transactions on, 40 (1995), 1626-1630.
doi: 10.1109/9.412634. |
| [15] |
W. L. Miranker,
The wave equation in a medium in motion, IBM Journal of Research and Development, 4 (1960), 36-42.
doi: 10.1147/rd.41.0036. |
| [16] |
S. Nicaise and C. Pignotti,
Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, Siam Journal on Control Optimization, 45 (2006), 1561-1585.
doi: 10.1137/060648891. |
| [17] |
S. Nicaise,
Interior feedback stabilization of wave equations with time dependent delay, Electronic Journal of Differential Equations, 41 (2011), 1-20.
doi: 10.1016/j.sysconle.2011.09.016. |
| [18] |
S. Nicaise and J. Valein,
Stabilization of the wave equation on 1-d networks with a delay term in the nodal feedbacks, Networks and Heterogeneous Midea, 2 (2007), 425-479.
doi: 10.3934/nhm.2007.2.425. |
| [19] |
S. Nicaise and J. Valein,
Stability of the heat and of the wave equations with boundary time-varying delays, Discrete and Continuous Dynamical Systems-Series S, 2 (2009), 559-581.
doi: 10.3934/dcdss.2009.2.559. |
| [20] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. |
| [21] |
C. Pignotti,
A note on stabilization of locally damped wave equations with time delay, Systems Control Letters, 61 (2012), 92-97.
doi: 10.1016/j.sysconle.2011.09.016. |
| [22] |
I. Suh and Z. Bien,
Use of time-delay actions in the controller design, IEEE Transactions on Automatic Control, 25 (1980), 600-603.
|
| [23] |
Y. Shang and G. Q. Xu,
The stability of a wave equation with delay-dependent position, IMA Journal of Mathematical Control and Information, 28 (2011), 75-95.
doi: 10.1093/imamci/dnq026. |
| [24] |
H. Wang and G. Xu,
Exponential stabilization of 1-d wave equation with input delay, Wseas Transactions on Mathematics, 12 (2013), 1001-1013.
|
| [25] |
G. Q. Xu, S. Yung and L. Li,
Stabilization of wave systems with input delay in the boundary control, Control, Optimisation and Calculus of Variations, 12 (2006), 770-785.
doi: 10.1051/cocv:2006021. |
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