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Finite-time stabilization of a network of strings
1. | Institut Elie Cartan de Lorraine, UMR-CNRS 7502, Université de Lorraine, Ile du Saulcy, 57045 Metz Cedex 1 |
2. | Laboratoire de Mathématiques et Physique Théorique, Université de Tours, UFR Sciences et Techniques, Parc de Grandmont, 37200 Tours, France |
3. | Centre Automatique et Systèmes, MINES ParisTech, PSL Research University, 60 Boulevard Saint-Michel, 75272 Paris Cedex 06, France |
References:
[1] |
K. Ammari and M. Jellouli, Stabilization of star-shaped networks of strings, Differential and Integral Equations, 17 (2004), 1395-1410. |
[2] |
K. Ammari, M. Jellouli and M. Khenissi, Stabilization of generic trees of strings, J. Dyn. Control Syst., 11 (2005), 177-193.
doi: 10.1007/s10883-005-4169-7. |
[3] |
A. Bacciotti and L. Rosier, Liapunov Functions and Stability in Control Theory, $2^{nd}$ edition, Communications and Control Engineering Series, Springer-Verlag, Berlin, 2005.
doi: 10.1007/b139028. |
[4] |
S. P. Bhat and D. S. Bernstein, Finite-time stability of continuous autonomous systems, SIAM J. Control Optim., 38 (2000), 751-766.
doi: 10.1137/S0363012997321358. |
[5] |
S. Cox and E. Zuazua, The rate at which energy decays in a string damped at one end, Indiana Univ. Math. J., 44 (1995), 545-573.
doi: 10.1512/iumj.1995.44.2001. |
[6] |
R. Dáger and E. Zuazua, Controllability of tree-shaped networks of vibrating strings, C. R. Acad. Sci. Paris Sér. I Math., 332 (2001), 1087-1092.
doi: 10.1016/S0764-4442(01)01942-5. |
[7] |
R. Dáger and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-Structures, Mathématiques & Applications (Berlin) [Mathematics & Applications], 50, Springer-Verlag, Berlin, 2006.
doi: 10.1007/3-540-37726-3. |
[8] |
M. Gugat, Optimal boundary feedback stabilization of a string with moving boundary, IMA J. Math. Control Inform., 25 (2008), 111-121.
doi: 10.1093/imamci/dnm014. |
[9] |
M. Gugat, M. Dick and G. Leugering, Gas flow in fan-shaped networks: Classical solutions and feedback stabilization, SIAM J. Control Optim., 49 (2011), 2101-2117.
doi: 10.1137/100799824. |
[10] |
M. Gugat and M. Sigalotti, Stars of vibrating strings: Switching boundary feedback stabilization, Netw. Heterog. Media, 5 (2010), 299-314.
doi: 10.3934/nhm.2010.5.299. |
[11] |
V. T Haimo, Finite time controllers, SIAM J. Control Optim., 24 (1986), 760-770.
doi: 10.1137/0324047. |
[12] |
V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM: Research in Applied Mathematics, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. |
[13] |
J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modeling, analysis and control of dynamic elastic multi-link structures, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1994.
doi: 10.1007/978-1-4612-0273-8. |
[14] |
A. Majda, Disappearing solutions for the dissipative wave equation,, Indiana Univ. Math. J., 24 (): 1119.
|
[15] |
E. Moulay and W. Perruquetti, Finite-time stability and stabilization: state of the art, in Advances in Variable Structure and Sliding Mode Control, Lecture Notes in Control and Inform. Sci., 334, Springer, Berlin, 2006, 23-41.
doi: 10.1007/11612735_2. |
[16] |
V. Perrollaz and L. Rosier, Finite-time stabilization of hyperbolic systems over a bounded interval, in 1st IFAC workshop on Control of Systems Governed by Partial Differential Equations (CPDE2013), 2013, 239-244. |
[17] |
V. Perrollaz and L. Rosier, Finite-time stabilization of $2\times 2$ hyperbolic systems on tree-shaped networks, SIAM J. Control Optim., 52 (2014), 143-163.
doi: 10.1137/130910762. |
[18] |
Y. Shang, D. Liu and G. Xu, Super-stability and the spectrum of one-dimensional wave equations on general feedback controlled networks, IMA J. Math. Control and Inform., 31 (2014), 73-99.
doi: 10.1093/imamci/dnt003. |
[19] |
J. Valein and E. Zuazua, Stabilization of the wave equation on 1-D networks, SIAM J. Control Optim., 48 (2009), 2771-2797.
doi: 10.1137/080733590. |
[20] |
Y. Zhang and G. Xu, Controller design for bush-type 1-D wave networks, ESAIM Control Optim. Calc. Var., 18 (2012), 208-228.
doi: 10.1051/cocv/2010050. |
[21] |
Y. Zhang and G. Xu, Exponential and super stability of a wave network, Acta Appl. Math., 124 (2013), 19-41.
doi: 10.1007/s10440-012-9768-1. |
[22] |
G. Q. Xu, D. Y. Liu and Y. Q. Liu, Abstract second order hyperbolic system and applications to controlled network of strings, SIAM J. Control Optim., 47 (2008), 1762-1784.
doi: 10.1137/060649367. |
show all references
References:
[1] |
K. Ammari and M. Jellouli, Stabilization of star-shaped networks of strings, Differential and Integral Equations, 17 (2004), 1395-1410. |
[2] |
K. Ammari, M. Jellouli and M. Khenissi, Stabilization of generic trees of strings, J. Dyn. Control Syst., 11 (2005), 177-193.
doi: 10.1007/s10883-005-4169-7. |
[3] |
A. Bacciotti and L. Rosier, Liapunov Functions and Stability in Control Theory, $2^{nd}$ edition, Communications and Control Engineering Series, Springer-Verlag, Berlin, 2005.
doi: 10.1007/b139028. |
[4] |
S. P. Bhat and D. S. Bernstein, Finite-time stability of continuous autonomous systems, SIAM J. Control Optim., 38 (2000), 751-766.
doi: 10.1137/S0363012997321358. |
[5] |
S. Cox and E. Zuazua, The rate at which energy decays in a string damped at one end, Indiana Univ. Math. J., 44 (1995), 545-573.
doi: 10.1512/iumj.1995.44.2001. |
[6] |
R. Dáger and E. Zuazua, Controllability of tree-shaped networks of vibrating strings, C. R. Acad. Sci. Paris Sér. I Math., 332 (2001), 1087-1092.
doi: 10.1016/S0764-4442(01)01942-5. |
[7] |
R. Dáger and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-Structures, Mathématiques & Applications (Berlin) [Mathematics & Applications], 50, Springer-Verlag, Berlin, 2006.
doi: 10.1007/3-540-37726-3. |
[8] |
M. Gugat, Optimal boundary feedback stabilization of a string with moving boundary, IMA J. Math. Control Inform., 25 (2008), 111-121.
doi: 10.1093/imamci/dnm014. |
[9] |
M. Gugat, M. Dick and G. Leugering, Gas flow in fan-shaped networks: Classical solutions and feedback stabilization, SIAM J. Control Optim., 49 (2011), 2101-2117.
doi: 10.1137/100799824. |
[10] |
M. Gugat and M. Sigalotti, Stars of vibrating strings: Switching boundary feedback stabilization, Netw. Heterog. Media, 5 (2010), 299-314.
doi: 10.3934/nhm.2010.5.299. |
[11] |
V. T Haimo, Finite time controllers, SIAM J. Control Optim., 24 (1986), 760-770.
doi: 10.1137/0324047. |
[12] |
V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM: Research in Applied Mathematics, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. |
[13] |
J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modeling, analysis and control of dynamic elastic multi-link structures, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1994.
doi: 10.1007/978-1-4612-0273-8. |
[14] |
A. Majda, Disappearing solutions for the dissipative wave equation,, Indiana Univ. Math. J., 24 (): 1119.
|
[15] |
E. Moulay and W. Perruquetti, Finite-time stability and stabilization: state of the art, in Advances in Variable Structure and Sliding Mode Control, Lecture Notes in Control and Inform. Sci., 334, Springer, Berlin, 2006, 23-41.
doi: 10.1007/11612735_2. |
[16] |
V. Perrollaz and L. Rosier, Finite-time stabilization of hyperbolic systems over a bounded interval, in 1st IFAC workshop on Control of Systems Governed by Partial Differential Equations (CPDE2013), 2013, 239-244. |
[17] |
V. Perrollaz and L. Rosier, Finite-time stabilization of $2\times 2$ hyperbolic systems on tree-shaped networks, SIAM J. Control Optim., 52 (2014), 143-163.
doi: 10.1137/130910762. |
[18] |
Y. Shang, D. Liu and G. Xu, Super-stability and the spectrum of one-dimensional wave equations on general feedback controlled networks, IMA J. Math. Control and Inform., 31 (2014), 73-99.
doi: 10.1093/imamci/dnt003. |
[19] |
J. Valein and E. Zuazua, Stabilization of the wave equation on 1-D networks, SIAM J. Control Optim., 48 (2009), 2771-2797.
doi: 10.1137/080733590. |
[20] |
Y. Zhang and G. Xu, Controller design for bush-type 1-D wave networks, ESAIM Control Optim. Calc. Var., 18 (2012), 208-228.
doi: 10.1051/cocv/2010050. |
[21] |
Y. Zhang and G. Xu, Exponential and super stability of a wave network, Acta Appl. Math., 124 (2013), 19-41.
doi: 10.1007/s10440-012-9768-1. |
[22] |
G. Q. Xu, D. Y. Liu and Y. Q. Liu, Abstract second order hyperbolic system and applications to controlled network of strings, SIAM J. Control Optim., 47 (2008), 1762-1784.
doi: 10.1137/060649367. |
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