September  2016, 11(3): 527-543. doi: 10.3934/nhm.2016008

The exponential decay rate of generic tree of 1-d wave equations with boundary feedback controls

1. 

Department of Mathematics, Tianjin University, Haihe Education Park, Tianjin, Tianjin, MO 300000, China, China

Received  October 2014 Revised  February 2016 Published  August 2016

In this paper, we study the exponential decay rate of generic tree of 1-d wave equations with boundary feedback controls. For the networks, there are some results on the exponential stability, but no result on estimate of the decay rate. The present work mainly estimates the decay rate for these systems, including signal wave equation, serially connected wave equations, and generic tree of 1-d wave equations. By defining the weighted energy functional of the system, and choosing suitable weighted functions, we obtain the estimation value of decay rate of the systems.
Citation: Yaru Xie, Genqi Xu. The exponential decay rate of generic tree of 1-d wave equations with boundary feedback controls. Networks & Heterogeneous Media, 2016, 11 (3) : 527-543. doi: 10.3934/nhm.2016008
References:
[1]

K. Ammari and M. Jellouli, Stabilization of star-shaped tree of elastic strings,, Differential & Integral Equations, 17 (2004), 1395. Google Scholar

[2]

K. Ammari, M. Jellouli and M. Khenissi, Stabilization of generic trees of strings,, Journal of Dynamical & Control Systems, 11 (2005), 177. doi: 10.1007/s10883-005-4169-7. Google Scholar

[3]

K. Ammari and D. Mercier, Boundary feedback stabilization of a chain of serially connected strings,, Evol. Equ. Control Theory, 4 (2015), 1. doi: 10.3934/eect.2015.4.1. Google Scholar

[4]

K. Ammari, D. Mercier and V. Régnier, Spectral analysis of the Schrodinger operator on binary tree-shaped networks and applications,, Journal of Differential Equations, 259 (2015), 6923. doi: 10.1016/j.jde.2015.08.017. Google Scholar

[5]

K. Ammari, A. Henrot and M. Tucsnak, Asymptotic behaviour of the solutions and optimal location of the actuator for the pointwise stabilization of a string,, Asymptotic Analysis, 28 (2001), 215. Google Scholar

[6]

K. Bartecki, A general transfer function representation for a class of hyperbolic distributed parameter systems,, International Journal of Applied Mathematics & Computer Science, 23 (2013), 291. doi: 10.2478/amcs-2013-0022. Google Scholar

[7]

A. V. Balakrishnan, On superstable semigroup of operators,, Dynamic Systems & Applications, 5 (1996), 371. Google Scholar

[8]

J. M. Coron, B. D'Andrea-Novel and G. Bastin, A lyapunov approach to control irrigation canals modeled by the Saint Venant equations,, Control Conference. IEEE, (1999). Google Scholar

[9]

J. M. Coron, J. De Halleux and G. Bastin, On boundary control design for quasi-linear hyperbolic systems with entropies as Lyapunov functions,, Proceedings of the IEEE Conference on Decision and Control, 3 (2003), 3010. Google Scholar

[10]

J. M. Coron, B. d'Andrea-Novel and G. Bastin, A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws,, Proceedings of the IEEE Conference on Decision & Control, 52 (2007), 2. doi: 10.1109/TAC.2006.887903. Google Scholar

[11]

R. F. Curtain and H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory,, Springer-Verlag, (1995). doi: 10.1007/978-1-4612-4224-6. Google Scholar

[12]

S. Cox and E. Zuazua, The rate at which energy decays in a damped string,, Communication Partial Differential Equations, 19 (1994), 213. doi: 10.1080/03605309408821015. Google Scholar

[13]

R. Dager and E. Zuazua, Controllability of star-shaped networks of strings,, C. R. Acad. Sci. Paris, 332 (2001), 621. doi: 10.1016/S0764-4442(01)01876-6. Google Scholar

[14]

R. Dager, Observation and control of vibrations in tree-shaped networks of strings,, SIAM J. Control & Optim, 43 (2004), 590. doi: 10.1137/S0363012903421844. Google Scholar

[15]

R. Dager and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-Structures,, Springer Berlin, (2006). doi: 10.1007/3-540-37726-3. Google Scholar

[16]

A. Diagne, G. Bastin and J. M. Coron, Lyapunov exponential stability of 1-D linear hyperbolic systems of balance laws,, Automatica, 48 (2012), 109. doi: 10.1016/j.automatica.2011.09.030. Google Scholar

[17]

Y. N. Guo and G. Q. Xu, Stability and Riesz basis property for general network of strings,, Journal of dynamical & control systems, 15 (2009), 223. doi: 10.1007/s10883-009-9064-1. Google Scholar

[18]

Y. N. Guo and G. Q. Xu, Exponential stabilization of a tree shaped network of strings with variable coefficients,, Glasgow Mathematical Journal, 53 (2011), 481. doi: 10.1017/S0017089511000085. Google Scholar

[19]

Y. N. Guo and G. Q. Xu, Exponential stabilization of variable coefficient wave equations in a generic tree with small time-delays in the nodal feedbacks,, Journal of Mathematical Analysis & Applications, 395 (2012), 727. doi: 10.1016/j.jmaa.2012.05.079. Google Scholar

[20]

B. Z. Guo and Y. Xie, A sufficient condition on Riesz basis with parentheses of non-selfadjoint operator and application to a serially connected string system under joint feedbacks,, SIAM journal on control & optimization, 43 (2004), 1234. doi: 10.1137/S0363012902420352. Google Scholar

[21]

M. Gugat and M. Herty, Existence of classical solutions and feedback stabilization for the flow in gas networks,, ESAIM Control Optimisation & Calculus of Variations, 17 (2011), 28. doi: 10.1051/cocv/2009035. Google Scholar

[22]

F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces,, Ann. Differential Equations, 1 (1985), 43. Google Scholar

[23]

Z. J. Han and L. Wang, Riesz basis property and stability of planar networks of controlled strings,, Acta Applicandae Mathematicae, 110 (2010), 511. doi: 10.1007/s10440-009-9459-8. Google Scholar

[24]

Z. J. Han and G. Q. Xu, Spectrum and dynamical behavior of a kind of planar network of non-uniform strings with non-collocated feedbacks,, Networks & Heterogeneous Media, 5 (2010), 315. doi: 10.3934/nhm.2010.5.315. Google Scholar

[25]

Z. J. Han and G. Q. Xu, Stabilization and SDG condition of serially connected vibrating strings system with discontinuous displacement,, Asian Journal of Control, 14 (2012), 95. doi: 10.1002/asjc.218. Google Scholar

[26]

M. Jellouli, Spectral analysis for a degenerate tree and applications,, International Journal of Control, 88 (2015), 1647. doi: 10.1080/00207179.2015.1012652. Google Scholar

[27]

M.Krstic, B. Z. Guo and A. Balogh, Output-feedback stabilization of an unstable wave equation,, Automatica, 44 (2008), 63. doi: 10.1016/j.automatica.2007.05.012. Google Scholar

[28]

J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modelling, Analysis and Control of Dynamic Elastic Multi-Link Structures,, Systems & Control: Foundations & Applications, (1994). doi: 10.1007/978-1-4612-0273-8. Google Scholar

[29]

J. E. Lagnese, Recent progress and open problems in control of multi-link elastic structures,, Contemp. Math, 209 (1997), 161. doi: 10.1090/conm/209/02765. Google Scholar

[30]

J. L. Lions, Exact controllability, stabilization and perturbations for distributed parameter system,, SIAM Review, 30 (1988), 1. doi: 10.1137/1030001. Google Scholar

[31]

K. S. Liu, F. L. Huang and G. Chen, Exponential stability analysis of a long chain of coupled vibrating strings with dissipative linkage,, SIAM Journal on Applied Mathematics, 49 (1989), 1694. doi: 10.1137/0149102. Google Scholar

[32]

G. Leugering and E. Zuazua, On exact controllability of generic trees,, Esaim Proceedings, 8 (2000), 95. doi: 10.1051/proc:2000007. Google Scholar

[33]

G. Leugering, Dynamic domain decomposition of optimal control problems for networks of strings and Timoshenko beams,, SIAM Journal on Control & Optimization, 37 (1999), 1649. doi: 10.1137/S0363012997331986. Google Scholar

[34]

D. Y. Liu, Y. F. Shang and G. Q. Xu, Design of controllers and compensators of a serially connected string system and its Riesz basis,, Kongzhi Lilun Yu Yinyong/Control Theory & Applications, 5 (2008), 815. Google Scholar

[35]

M. Najafi, G. R. Sarhangi and H. Wang, Stabilizability of coupled wave equations in parallel under various boundary conditions,, Automatic Control IEEE Transactions on, 42 (1997), 1308. doi: 10.1109/9.623099. Google Scholar

[36]

S. Nicaise and J. Valein, Stabilization of the wave equation on 1-D networks with a delay-term in the feedbacks,, Networks & Heterogeneous Media, 2 (2007), 425. doi: 10.3934/nhm.2007.2.425. Google Scholar

[37]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Berlin, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar

[38]

D. L. Rusell, Mathematical models for the elastic beam and their control-theoretic implications,, Semigroups, 152 (1986), 177. Google Scholar

[39]

S. Rolewicz, On controllability of systems of strings, Studia Math,, Studia Mathematica, 36 (1970), 105. Google Scholar

[40]

Y. F. Shang, D. Y. Liu and G. Q. Xu, Super-stability and the spectrum of one-dimensional wave equations on general feedback controlled networks,, IMA Journal of Mathematical Control & Information, 31 (2014), 73. doi: 10.1093/imamci/dnt003. Google Scholar

[41]

M. A. Shubov, Basis property of eigenfunctions of nonselfadjoint operator pencils generated by the equations of nonhomogeneous damped string,, Integral Equations & Operator Theory, 25 (1996), 289. doi: 10.1007/BF01262296. Google Scholar

[42]

E. J. P. G. Schmidt, On the modelling and exact controllability of networks of vibrating strings,, Siam Journal on Control & Optimization, 30 (1992), 229. doi: 10.1137/0330015. Google Scholar

[43]

L. Wang, Z. J. Han and G. Q. Xu, Exponential stability of serially connected thermoelastic system of type II with nodal damping,, Applicable Analysis, 93 (2014), 1495. doi: 10.1080/00036811.2013.836596. Google Scholar

[44]

H. Wang and G. Q. Xu, Exponential stabilization of 1-d wave equation with input delay,, Wseas Transactions on Mathematics, 10 (2013), 1001. Google Scholar

[45]

G. Q. Xu, S. P. Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control,, ESAIM Control Optimisation & Calculus of Variations, 12 (2006), 770. doi: 10.1051/cocv:2006021. Google Scholar

[46]

G. Q. Xu, Stabilization of string system with linear boundary feedback,, Nonlinear Analysis Hybrid Systems, 1 (2007), 383. doi: 10.1016/j.nahs.2006.07.003. Google Scholar

[47]

G. Q. Xu and B. Z. Guo, Riesz basis property of evolution equations in Hilbert spaces and application to a coupled string equation,, SIAM Journal on Control & Optimization, 42 (2003), 966. doi: 10.1137/S0363012901400081. Google Scholar

[48]

G. Q. Xu, D. Y. Liu and Y. Q. Liu, Abstract second order hyperbolic system and applications to controlled network of strings,, SIAM Journal on Control & Optimization, 47 (2008), 1762. doi: 10.1137/060649367. Google Scholar

[49]

G. Q. Xu and N. E. Mastorakis, Differential Equations on Metric Graph,, Athens: World Scientific and Engineering Academy and Society, (2010). Google Scholar

[50]

G. Q. Xu and Y. X. Zhang, The exponential stability of complex differential networks,, Journal of Systems Science & Mathematical Sciences, 29 (2009), 1399. Google Scholar

[51]

E. Zuazua, Control and stabilization of waves on 1-d networks,, Modelling and Optimisation of Flows on Networks, 2062 (2013), 463. doi: 10.1007/978-3-642-32160-3_9. Google Scholar

[52]

Y. X. Zhang and G. Q. Xu, Controller design for Bush-type 1-D wave neworks,, ESAIM Control Optimisation & Calculus of Variations, 18 (2012), 208. doi: 10.1051/cocv/2010050. Google Scholar

[53]

Y. X. Zhang and G. Q. Xu, A new approach for the stability analysis of wave networks,, Abstract and Applied Analysis, 2014 (2014). doi: 10.1155/2014/724512. Google Scholar

[54]

Y. X. Zhang and G. Q. Xu, Exponential and super stability of a wave network,, Acta Applicandae Mathematicae An International Survey Journal on Applying Mathematics & Mathematical Applications, 124 (2013), 19. doi: 10.1007/s10440-012-9768-1. Google Scholar

show all references

References:
[1]

K. Ammari and M. Jellouli, Stabilization of star-shaped tree of elastic strings,, Differential & Integral Equations, 17 (2004), 1395. Google Scholar

[2]

K. Ammari, M. Jellouli and M. Khenissi, Stabilization of generic trees of strings,, Journal of Dynamical & Control Systems, 11 (2005), 177. doi: 10.1007/s10883-005-4169-7. Google Scholar

[3]

K. Ammari and D. Mercier, Boundary feedback stabilization of a chain of serially connected strings,, Evol. Equ. Control Theory, 4 (2015), 1. doi: 10.3934/eect.2015.4.1. Google Scholar

[4]

K. Ammari, D. Mercier and V. Régnier, Spectral analysis of the Schrodinger operator on binary tree-shaped networks and applications,, Journal of Differential Equations, 259 (2015), 6923. doi: 10.1016/j.jde.2015.08.017. Google Scholar

[5]

K. Ammari, A. Henrot and M. Tucsnak, Asymptotic behaviour of the solutions and optimal location of the actuator for the pointwise stabilization of a string,, Asymptotic Analysis, 28 (2001), 215. Google Scholar

[6]

K. Bartecki, A general transfer function representation for a class of hyperbolic distributed parameter systems,, International Journal of Applied Mathematics & Computer Science, 23 (2013), 291. doi: 10.2478/amcs-2013-0022. Google Scholar

[7]

A. V. Balakrishnan, On superstable semigroup of operators,, Dynamic Systems & Applications, 5 (1996), 371. Google Scholar

[8]

J. M. Coron, B. D'Andrea-Novel and G. Bastin, A lyapunov approach to control irrigation canals modeled by the Saint Venant equations,, Control Conference. IEEE, (1999). Google Scholar

[9]

J. M. Coron, J. De Halleux and G. Bastin, On boundary control design for quasi-linear hyperbolic systems with entropies as Lyapunov functions,, Proceedings of the IEEE Conference on Decision and Control, 3 (2003), 3010. Google Scholar

[10]

J. M. Coron, B. d'Andrea-Novel and G. Bastin, A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws,, Proceedings of the IEEE Conference on Decision & Control, 52 (2007), 2. doi: 10.1109/TAC.2006.887903. Google Scholar

[11]

R. F. Curtain and H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory,, Springer-Verlag, (1995). doi: 10.1007/978-1-4612-4224-6. Google Scholar

[12]

S. Cox and E. Zuazua, The rate at which energy decays in a damped string,, Communication Partial Differential Equations, 19 (1994), 213. doi: 10.1080/03605309408821015. Google Scholar

[13]

R. Dager and E. Zuazua, Controllability of star-shaped networks of strings,, C. R. Acad. Sci. Paris, 332 (2001), 621. doi: 10.1016/S0764-4442(01)01876-6. Google Scholar

[14]

R. Dager, Observation and control of vibrations in tree-shaped networks of strings,, SIAM J. Control & Optim, 43 (2004), 590. doi: 10.1137/S0363012903421844. Google Scholar

[15]

R. Dager and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-Structures,, Springer Berlin, (2006). doi: 10.1007/3-540-37726-3. Google Scholar

[16]

A. Diagne, G. Bastin and J. M. Coron, Lyapunov exponential stability of 1-D linear hyperbolic systems of balance laws,, Automatica, 48 (2012), 109. doi: 10.1016/j.automatica.2011.09.030. Google Scholar

[17]

Y. N. Guo and G. Q. Xu, Stability and Riesz basis property for general network of strings,, Journal of dynamical & control systems, 15 (2009), 223. doi: 10.1007/s10883-009-9064-1. Google Scholar

[18]

Y. N. Guo and G. Q. Xu, Exponential stabilization of a tree shaped network of strings with variable coefficients,, Glasgow Mathematical Journal, 53 (2011), 481. doi: 10.1017/S0017089511000085. Google Scholar

[19]

Y. N. Guo and G. Q. Xu, Exponential stabilization of variable coefficient wave equations in a generic tree with small time-delays in the nodal feedbacks,, Journal of Mathematical Analysis & Applications, 395 (2012), 727. doi: 10.1016/j.jmaa.2012.05.079. Google Scholar

[20]

B. Z. Guo and Y. Xie, A sufficient condition on Riesz basis with parentheses of non-selfadjoint operator and application to a serially connected string system under joint feedbacks,, SIAM journal on control & optimization, 43 (2004), 1234. doi: 10.1137/S0363012902420352. Google Scholar

[21]

M. Gugat and M. Herty, Existence of classical solutions and feedback stabilization for the flow in gas networks,, ESAIM Control Optimisation & Calculus of Variations, 17 (2011), 28. doi: 10.1051/cocv/2009035. Google Scholar

[22]

F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces,, Ann. Differential Equations, 1 (1985), 43. Google Scholar

[23]

Z. J. Han and L. Wang, Riesz basis property and stability of planar networks of controlled strings,, Acta Applicandae Mathematicae, 110 (2010), 511. doi: 10.1007/s10440-009-9459-8. Google Scholar

[24]

Z. J. Han and G. Q. Xu, Spectrum and dynamical behavior of a kind of planar network of non-uniform strings with non-collocated feedbacks,, Networks & Heterogeneous Media, 5 (2010), 315. doi: 10.3934/nhm.2010.5.315. Google Scholar

[25]

Z. J. Han and G. Q. Xu, Stabilization and SDG condition of serially connected vibrating strings system with discontinuous displacement,, Asian Journal of Control, 14 (2012), 95. doi: 10.1002/asjc.218. Google Scholar

[26]

M. Jellouli, Spectral analysis for a degenerate tree and applications,, International Journal of Control, 88 (2015), 1647. doi: 10.1080/00207179.2015.1012652. Google Scholar

[27]

M.Krstic, B. Z. Guo and A. Balogh, Output-feedback stabilization of an unstable wave equation,, Automatica, 44 (2008), 63. doi: 10.1016/j.automatica.2007.05.012. Google Scholar

[28]

J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modelling, Analysis and Control of Dynamic Elastic Multi-Link Structures,, Systems & Control: Foundations & Applications, (1994). doi: 10.1007/978-1-4612-0273-8. Google Scholar

[29]

J. E. Lagnese, Recent progress and open problems in control of multi-link elastic structures,, Contemp. Math, 209 (1997), 161. doi: 10.1090/conm/209/02765. Google Scholar

[30]

J. L. Lions, Exact controllability, stabilization and perturbations for distributed parameter system,, SIAM Review, 30 (1988), 1. doi: 10.1137/1030001. Google Scholar

[31]

K. S. Liu, F. L. Huang and G. Chen, Exponential stability analysis of a long chain of coupled vibrating strings with dissipative linkage,, SIAM Journal on Applied Mathematics, 49 (1989), 1694. doi: 10.1137/0149102. Google Scholar

[32]

G. Leugering and E. Zuazua, On exact controllability of generic trees,, Esaim Proceedings, 8 (2000), 95. doi: 10.1051/proc:2000007. Google Scholar

[33]

G. Leugering, Dynamic domain decomposition of optimal control problems for networks of strings and Timoshenko beams,, SIAM Journal on Control & Optimization, 37 (1999), 1649. doi: 10.1137/S0363012997331986. Google Scholar

[34]

D. Y. Liu, Y. F. Shang and G. Q. Xu, Design of controllers and compensators of a serially connected string system and its Riesz basis,, Kongzhi Lilun Yu Yinyong/Control Theory & Applications, 5 (2008), 815. Google Scholar

[35]

M. Najafi, G. R. Sarhangi and H. Wang, Stabilizability of coupled wave equations in parallel under various boundary conditions,, Automatic Control IEEE Transactions on, 42 (1997), 1308. doi: 10.1109/9.623099. Google Scholar

[36]

S. Nicaise and J. Valein, Stabilization of the wave equation on 1-D networks with a delay-term in the feedbacks,, Networks & Heterogeneous Media, 2 (2007), 425. doi: 10.3934/nhm.2007.2.425. Google Scholar

[37]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Berlin, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar

[38]

D. L. Rusell, Mathematical models for the elastic beam and their control-theoretic implications,, Semigroups, 152 (1986), 177. Google Scholar

[39]

S. Rolewicz, On controllability of systems of strings, Studia Math,, Studia Mathematica, 36 (1970), 105. Google Scholar

[40]

Y. F. Shang, D. Y. Liu and G. Q. Xu, Super-stability and the spectrum of one-dimensional wave equations on general feedback controlled networks,, IMA Journal of Mathematical Control & Information, 31 (2014), 73. doi: 10.1093/imamci/dnt003. Google Scholar

[41]

M. A. Shubov, Basis property of eigenfunctions of nonselfadjoint operator pencils generated by the equations of nonhomogeneous damped string,, Integral Equations & Operator Theory, 25 (1996), 289. doi: 10.1007/BF01262296. Google Scholar

[42]

E. J. P. G. Schmidt, On the modelling and exact controllability of networks of vibrating strings,, Siam Journal on Control & Optimization, 30 (1992), 229. doi: 10.1137/0330015. Google Scholar

[43]

L. Wang, Z. J. Han and G. Q. Xu, Exponential stability of serially connected thermoelastic system of type II with nodal damping,, Applicable Analysis, 93 (2014), 1495. doi: 10.1080/00036811.2013.836596. Google Scholar

[44]

H. Wang and G. Q. Xu, Exponential stabilization of 1-d wave equation with input delay,, Wseas Transactions on Mathematics, 10 (2013), 1001. Google Scholar

[45]

G. Q. Xu, S. P. Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control,, ESAIM Control Optimisation & Calculus of Variations, 12 (2006), 770. doi: 10.1051/cocv:2006021. Google Scholar

[46]

G. Q. Xu, Stabilization of string system with linear boundary feedback,, Nonlinear Analysis Hybrid Systems, 1 (2007), 383. doi: 10.1016/j.nahs.2006.07.003. Google Scholar

[47]

G. Q. Xu and B. Z. Guo, Riesz basis property of evolution equations in Hilbert spaces and application to a coupled string equation,, SIAM Journal on Control & Optimization, 42 (2003), 966. doi: 10.1137/S0363012901400081. Google Scholar

[48]

G. Q. Xu, D. Y. Liu and Y. Q. Liu, Abstract second order hyperbolic system and applications to controlled network of strings,, SIAM Journal on Control & Optimization, 47 (2008), 1762. doi: 10.1137/060649367. Google Scholar

[49]

G. Q. Xu and N. E. Mastorakis, Differential Equations on Metric Graph,, Athens: World Scientific and Engineering Academy and Society, (2010). Google Scholar

[50]

G. Q. Xu and Y. X. Zhang, The exponential stability of complex differential networks,, Journal of Systems Science & Mathematical Sciences, 29 (2009), 1399. Google Scholar

[51]

E. Zuazua, Control and stabilization of waves on 1-d networks,, Modelling and Optimisation of Flows on Networks, 2062 (2013), 463. doi: 10.1007/978-3-642-32160-3_9. Google Scholar

[52]

Y. X. Zhang and G. Q. Xu, Controller design for Bush-type 1-D wave neworks,, ESAIM Control Optimisation & Calculus of Variations, 18 (2012), 208. doi: 10.1051/cocv/2010050. Google Scholar

[53]

Y. X. Zhang and G. Q. Xu, A new approach for the stability analysis of wave networks,, Abstract and Applied Analysis, 2014 (2014). doi: 10.1155/2014/724512. Google Scholar

[54]

Y. X. Zhang and G. Q. Xu, Exponential and super stability of a wave network,, Acta Applicandae Mathematicae An International Survey Journal on Applying Mathematics & Mathematical Applications, 124 (2013), 19. doi: 10.1007/s10440-012-9768-1. Google Scholar

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