Exponential stability of 1-d wave equation with the boundary time delay based on the interior control

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June 2017, 10(3): 557-579. doi: 10.3934/dcdss.2017028

Exponential stability of 1-d wave equation with the boundary time delay based on the interior control

Yaru Xie and Genqi Xu

Department of Mathematics, Tianjin University, Haihe Education Park, Tianjin, Tianjin 300350, China

Received June 2016 Revised January 2017 Published February 2017

Fund Project: This work is supported by Science Foundation of China under Grant Nos.61174080, 61503275 and 61573252

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.

Keywords: Wave equation, Lyapunov function, boundary delay, exponential stability.

Mathematics Subject Classification: Primary: 35L05, 37L45; Secondary: 37L15.

Citation: Yaru Xie, Genqi Xu. Exponential stability of 1-d wave equation with the boundary time delay based on the interior control. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 557-579. doi: 10.3934/dcdss.2017028

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.

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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