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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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983.  Google Scholar

[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.  Google Scholar

[22]

I. Suh and Z. Bien, Use of time-delay actions in the controller design, IEEE Transactions on Automatic Control, 25 (1980), 600-603.   Google Scholar

[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.  Google Scholar

[24]

H. Wang and G. Xu, Exponential stabilization of 1-d wave equation with input delay, Wseas Transactions on Mathematics, 12 (2013), 1001-1013.   Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983.  Google Scholar

[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.  Google Scholar

[22]

I. Suh and Z. Bien, Use of time-delay actions in the controller design, IEEE Transactions on Automatic Control, 25 (1980), 600-603.   Google Scholar

[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.  Google Scholar

[24]

H. Wang and G. Xu, Exponential stabilization of 1-d wave equation with input delay, Wseas Transactions on Mathematics, 12 (2013), 1001-1013.   Google Scholar

[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.  Google Scholar

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