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Evolution of spoon-shaped networks
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 |
References:
[1] |
K. Ammari and M. Jellouli, Stabilization of star-shaped tree of elastic strings, Differential & Integral Equations, 17 (2004), 1395-1410. |
[2] |
K. Ammari, M. Jellouli and M. Khenissi, Stabilization of generic trees of strings, Journal of Dynamical & Control Systems, 11 (2005), 177-193.
doi: 10.1007/s10883-005-4169-7. |
[3] |
K. Ammari and D. Mercier, Boundary feedback stabilization of a chain of serially connected strings, Evol. Equ. Control Theory, 4 (2015), 1-19.
doi: 10.3934/eect.2015.4.1. |
[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), 6923-6959.
doi: 10.1016/j.jde.2015.08.017. |
[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), 215-240. |
[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), 291-307.
doi: 10.2478/amcs-2013-0022. |
[7] |
A. V. Balakrishnan, On superstable semigroup of operators, Dynamic Systems & Applications, 5 (1996), 371-384. |
[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. |
[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), 3010-3014. |
[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), 2-11.
doi: 10.1109/TAC.2006.887903. |
[11] |
R. F. Curtain and H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Springer-Verlag, New York, Berlin, Heidelberg, 1995.
doi: 10.1007/978-1-4612-4224-6. |
[12] |
S. Cox and E. Zuazua, The rate at which energy decays in a damped string, Communication Partial Differential Equations, 19 (1994), 213-243.
doi: 10.1080/03605309408821015. |
[13] |
R. Dager and E. Zuazua, Controllability of star-shaped networks of strings, C. R. Acad. Sci. Paris, Series I, 332 (2001), 621-626.
doi: 10.1016/S0764-4442(01)01876-6. |
[14] |
R. Dager, Observation and control of vibrations in tree-shaped networks of strings, SIAM J. Control & Optim, 43 (2004), 590-623.
doi: 10.1137/S0363012903421844. |
[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. |
[16] |
A. Diagne, G. Bastin and J. M. Coron, Lyapunov exponential stability of 1-D linear hyperbolic systems of balance laws, Automatica, 48 (2012), 109-114.
doi: 10.1016/j.automatica.2011.09.030. |
[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), 223-245.
doi: 10.1007/s10883-009-9064-1. |
[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), 481-499.
doi: 10.1017/S0017089511000085. |
[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), 727-746.
doi: 10.1016/j.jmaa.2012.05.079. |
[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), 1234-1252.
doi: 10.1137/S0363012902420352. |
[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), 28-51.
doi: 10.1051/cocv/2009035. |
[22] |
F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56. |
[23] |
Z. J. Han and L. Wang, Riesz basis property and stability of planar networks of controlled strings, Acta Applicandae Mathematicae, 110 (2010), 511-533.
doi: 10.1007/s10440-009-9459-8. |
[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), 315-334.
doi: 10.3934/nhm.2010.5.315. |
[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), 95-108.
doi: 10.1002/asjc.218. |
[26] |
M. Jellouli, Spectral analysis for a degenerate tree and applications, International Journal of Control, 88 (2015), 1647-1662.
doi: 10.1080/00207179.2015.1012652. |
[27] |
M.Krstic, B. Z. Guo and A. Balogh, Output-feedback stabilization of an unstable wave equation, Automatica, 44 (2008), 63-74.
doi: 10.1016/j.automatica.2007.05.012. |
[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. |
[29] |
J. E. Lagnese, Recent progress and open problems in control of multi-link elastic structures, Contemp. Math, 209 (1997), 161-175.
doi: 10.1090/conm/209/02765. |
[30] |
J. L. Lions, Exact controllability, stabilization and perturbations for distributed parameter system, SIAM Review, 30 (1988), 1-68.
doi: 10.1137/1030001. |
[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), 1694-1707.
doi: 10.1137/0149102. |
[32] |
G. Leugering and E. Zuazua, On exact controllability of generic trees, Esaim Proceedings, 8 (2000), 95-105.
doi: 10.1051/proc:2000007. |
[33] |
G. Leugering, Dynamic domain decomposition of optimal control problems for networks of strings and Timoshenko beams, SIAM Journal on Control & Optimization, 37 (1999), 1649-1675.
doi: 10.1137/S0363012997331986. |
[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), 815-818. |
[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), 1308-1312.
doi: 10.1109/9.623099. |
[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), 425-479.
doi: 10.3934/nhm.2007.2.425. |
[37] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Berlin, Germany: Springer-Verlag, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[38] |
D. L. Rusell, Mathematical models for the elastic beam and their control-theoretic implications, Semigroups, Theory and Applications, 152 (1986), 177-216. |
[39] |
S. Rolewicz, On controllability of systems of strings, Studia Math, Studia Mathematica, 36 (1970), 105-110. |
[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), 73-99.
doi: 10.1093/imamci/dnt003. |
[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), 289-328.
doi: 10.1007/BF01262296. |
[42] |
E. J. P. G. Schmidt, On the modelling and exact controllability of networks of vibrating strings, Siam Journal on Control & Optimization, 30 (1992), 229-245.
doi: 10.1137/0330015. |
[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), 1495-1514.
doi: 10.1080/00036811.2013.836596. |
[44] |
H. Wang and G. Q. Xu, Exponential stabilization of 1-d wave equation with input delay, Wseas Transactions on Mathematics, 10 (2013), 1001-1013. |
[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), 770-785.
doi: 10.1051/cocv:2006021. |
[46] |
G. Q. Xu, Stabilization of string system with linear boundary feedback, Nonlinear Analysis Hybrid Systems, 1 (2007), 383-397.
doi: 10.1016/j.nahs.2006.07.003. |
[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), 966-984.
doi: 10.1137/S0363012901400081. |
[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), 1762-1784.
doi: 10.1137/060649367. |
[49] |
G. Q. Xu and N. E. Mastorakis, Differential Equations on Metric Graph, Athens: World Scientific and Engineering Academy and Society, 2010. |
[50] |
G. Q. Xu and Y. X. Zhang, The exponential stability of complex differential networks, Journal of Systems Science & Mathematical Sciences, 29 (2009), 1399-1418. |
[51] |
E. Zuazua, Control and stabilization of waves on 1-d networks, Modelling and Optimisation of Flows on Networks, Springer Berlin Heidelberg, 2062 (2013), 463-493.
doi: 10.1007/978-3-642-32160-3_9. |
[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), 208-228.
doi: 10.1051/cocv/2010050. |
[53] |
Y. X. Zhang and G. Q. Xu, A new approach for the stability analysis of wave networks, Abstract and Applied Analysis, 2014 (2014), Art. ID 724512, 10 pp.
doi: 10.1155/2014/724512. |
[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), 19-41.
doi: 10.1007/s10440-012-9768-1. |
show all references
References:
[1] |
K. Ammari and M. Jellouli, Stabilization of star-shaped tree of elastic strings, Differential & Integral Equations, 17 (2004), 1395-1410. |
[2] |
K. Ammari, M. Jellouli and M. Khenissi, Stabilization of generic trees of strings, Journal of Dynamical & Control Systems, 11 (2005), 177-193.
doi: 10.1007/s10883-005-4169-7. |
[3] |
K. Ammari and D. Mercier, Boundary feedback stabilization of a chain of serially connected strings, Evol. Equ. Control Theory, 4 (2015), 1-19.
doi: 10.3934/eect.2015.4.1. |
[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), 6923-6959.
doi: 10.1016/j.jde.2015.08.017. |
[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), 215-240. |
[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), 291-307.
doi: 10.2478/amcs-2013-0022. |
[7] |
A. V. Balakrishnan, On superstable semigroup of operators, Dynamic Systems & Applications, 5 (1996), 371-384. |
[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. |
[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), 3010-3014. |
[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), 2-11.
doi: 10.1109/TAC.2006.887903. |
[11] |
R. F. Curtain and H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Springer-Verlag, New York, Berlin, Heidelberg, 1995.
doi: 10.1007/978-1-4612-4224-6. |
[12] |
S. Cox and E. Zuazua, The rate at which energy decays in a damped string, Communication Partial Differential Equations, 19 (1994), 213-243.
doi: 10.1080/03605309408821015. |
[13] |
R. Dager and E. Zuazua, Controllability of star-shaped networks of strings, C. R. Acad. Sci. Paris, Series I, 332 (2001), 621-626.
doi: 10.1016/S0764-4442(01)01876-6. |
[14] |
R. Dager, Observation and control of vibrations in tree-shaped networks of strings, SIAM J. Control & Optim, 43 (2004), 590-623.
doi: 10.1137/S0363012903421844. |
[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. |
[16] |
A. Diagne, G. Bastin and J. M. Coron, Lyapunov exponential stability of 1-D linear hyperbolic systems of balance laws, Automatica, 48 (2012), 109-114.
doi: 10.1016/j.automatica.2011.09.030. |
[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), 223-245.
doi: 10.1007/s10883-009-9064-1. |
[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), 481-499.
doi: 10.1017/S0017089511000085. |
[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), 727-746.
doi: 10.1016/j.jmaa.2012.05.079. |
[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), 1234-1252.
doi: 10.1137/S0363012902420352. |
[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), 28-51.
doi: 10.1051/cocv/2009035. |
[22] |
F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56. |
[23] |
Z. J. Han and L. Wang, Riesz basis property and stability of planar networks of controlled strings, Acta Applicandae Mathematicae, 110 (2010), 511-533.
doi: 10.1007/s10440-009-9459-8. |
[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), 315-334.
doi: 10.3934/nhm.2010.5.315. |
[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), 95-108.
doi: 10.1002/asjc.218. |
[26] |
M. Jellouli, Spectral analysis for a degenerate tree and applications, International Journal of Control, 88 (2015), 1647-1662.
doi: 10.1080/00207179.2015.1012652. |
[27] |
M.Krstic, B. Z. Guo and A. Balogh, Output-feedback stabilization of an unstable wave equation, Automatica, 44 (2008), 63-74.
doi: 10.1016/j.automatica.2007.05.012. |
[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. |
[29] |
J. E. Lagnese, Recent progress and open problems in control of multi-link elastic structures, Contemp. Math, 209 (1997), 161-175.
doi: 10.1090/conm/209/02765. |
[30] |
J. L. Lions, Exact controllability, stabilization and perturbations for distributed parameter system, SIAM Review, 30 (1988), 1-68.
doi: 10.1137/1030001. |
[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), 1694-1707.
doi: 10.1137/0149102. |
[32] |
G. Leugering and E. Zuazua, On exact controllability of generic trees, Esaim Proceedings, 8 (2000), 95-105.
doi: 10.1051/proc:2000007. |
[33] |
G. Leugering, Dynamic domain decomposition of optimal control problems for networks of strings and Timoshenko beams, SIAM Journal on Control & Optimization, 37 (1999), 1649-1675.
doi: 10.1137/S0363012997331986. |
[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), 815-818. |
[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), 1308-1312.
doi: 10.1109/9.623099. |
[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), 425-479.
doi: 10.3934/nhm.2007.2.425. |
[37] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Berlin, Germany: Springer-Verlag, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[38] |
D. L. Rusell, Mathematical models for the elastic beam and their control-theoretic implications, Semigroups, Theory and Applications, 152 (1986), 177-216. |
[39] |
S. Rolewicz, On controllability of systems of strings, Studia Math, Studia Mathematica, 36 (1970), 105-110. |
[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), 73-99.
doi: 10.1093/imamci/dnt003. |
[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), 289-328.
doi: 10.1007/BF01262296. |
[42] |
E. J. P. G. Schmidt, On the modelling and exact controllability of networks of vibrating strings, Siam Journal on Control & Optimization, 30 (1992), 229-245.
doi: 10.1137/0330015. |
[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), 1495-1514.
doi: 10.1080/00036811.2013.836596. |
[44] |
H. Wang and G. Q. Xu, Exponential stabilization of 1-d wave equation with input delay, Wseas Transactions on Mathematics, 10 (2013), 1001-1013. |
[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), 770-785.
doi: 10.1051/cocv:2006021. |
[46] |
G. Q. Xu, Stabilization of string system with linear boundary feedback, Nonlinear Analysis Hybrid Systems, 1 (2007), 383-397.
doi: 10.1016/j.nahs.2006.07.003. |
[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), 966-984.
doi: 10.1137/S0363012901400081. |
[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), 1762-1784.
doi: 10.1137/060649367. |
[49] |
G. Q. Xu and N. E. Mastorakis, Differential Equations on Metric Graph, Athens: World Scientific and Engineering Academy and Society, 2010. |
[50] |
G. Q. Xu and Y. X. Zhang, The exponential stability of complex differential networks, Journal of Systems Science & Mathematical Sciences, 29 (2009), 1399-1418. |
[51] |
E. Zuazua, Control and stabilization of waves on 1-d networks, Modelling and Optimisation of Flows on Networks, Springer Berlin Heidelberg, 2062 (2013), 463-493.
doi: 10.1007/978-3-642-32160-3_9. |
[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), 208-228.
doi: 10.1051/cocv/2010050. |
[53] |
Y. X. Zhang and G. Q. Xu, A new approach for the stability analysis of wave networks, Abstract and Applied Analysis, 2014 (2014), Art. ID 724512, 10 pp.
doi: 10.1155/2014/724512. |
[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), 19-41.
doi: 10.1007/s10440-012-9768-1. |
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