Finite-time stabilization of a network of strings

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December 2015, 5(4): 721-742. doi: 10.3934/mcrf.2015.5.721

Finite-time stabilization of a network of strings

Fatiha Alabau-Boussouira ^1, , Vincent Perrollaz ^2, and Lionel Rosier ^3,

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

Received October 2014 Revised January 2015 Published October 2015

Abstract
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We investigate the finite-time stabilization of a tree-shaped network of strings. Transparent boundary conditions are applied at all the external nodes. At any internal node, in addition to the usual continuity conditions, a modified Kirchhoff law incorporating a damping term $\alpha u_t$ with a coefficient $\alpha$ that may depend on the node is considered. We show that for a convenient choice of the sequence of coefficients $\alpha$, any solution of the wave equation on the network becomes constant after a finite time. The condition on the coefficients proves to be sharp at least for a star-shaped tree. Similar results are derived when we replace the transparent boundary condition by the Dirichlet (resp. Neumann) boundary condition at one external node. Our results lead to the finite-time stabilization even though the systems may not be dissipative.

Keywords: Kirchhoff law, wave equation, Finite-time stabilization, Riemann invariant., transparent boundary condition, network.

Mathematics Subject Classification: Primary: 93D15; Secondary: 34B45, 35L0.

Citation: Fatiha Alabau-Boussouira, Vincent Perrollaz, Lionel Rosier. Finite-time stabilization of a network of strings. Mathematical Control and Related Fields, 2015, 5 (4) : 721-742. doi: 10.3934/mcrf.2015.5.721

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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