# American Institute of Mathematical Sciences

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

## 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

Received  October 2014 Revised  January 2015 Published  October 2015

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.
Citation: Fatiha Alabau-Boussouira, Vincent Perrollaz, Lionel Rosier. Finite-time stabilization of a network of strings. Mathematical Control & 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. Google Scholar [2] K. Ammari, M. Jellouli and M. Khenissi, Stabilization of generic trees of strings,, J. Dyn. Control Syst., 11 (2005), 177. doi: 10.1007/s10883-005-4169-7. Google Scholar [3] A. Bacciotti and L. Rosier, Liapunov Functions and Stability in Control Theory,, $2^{nd}$ edition, (2005). doi: 10.1007/b139028. Google Scholar [4] S. P. Bhat and D. S. Bernstein, Finite-time stability of continuous autonomous systems,, SIAM J. Control Optim., 38 (2000), 751. doi: 10.1137/S0363012997321358. Google Scholar [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. doi: 10.1512/iumj.1995.44.2001. Google Scholar [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. doi: 10.1016/S0764-4442(01)01942-5. Google Scholar [7] R. Dáger and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-Structures,, Mathématiques & Applications (Berlin) [Mathematics & Applications], (2006). doi: 10.1007/3-540-37726-3. Google Scholar [8] M. Gugat, Optimal boundary feedback stabilization of a string with moving boundary,, IMA J. Math. Control Inform., 25 (2008), 111. doi: 10.1093/imamci/dnm014. Google Scholar [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. doi: 10.1137/100799824. Google Scholar [10] M. Gugat and M. Sigalotti, Stars of vibrating strings: Switching boundary feedback stabilization,, Netw. Heterog. Media, 5 (2010), 299. doi: 10.3934/nhm.2010.5.299. Google Scholar [11] V. T Haimo, Finite time controllers,, SIAM J. Control Optim., 24 (1986), 760. doi: 10.1137/0324047. Google Scholar [12] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method,, RAM: Research in Applied Mathematics, (1994). Google Scholar [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, (1994). doi: 10.1007/978-1-4612-0273-8. Google Scholar [14] A. Majda, Disappearing solutions for the dissipative wave equation,, Indiana Univ. Math. J., 24 (): 1119. Google Scholar [15] E. Moulay and W. Perruquetti, Finite-time stability and stabilization: state of the art,, in Advances in Variable Structure and Sliding Mode Control, (2006), 23. doi: 10.1007/11612735_2. Google Scholar [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. Google Scholar [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. doi: 10.1137/130910762. Google Scholar [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. doi: 10.1093/imamci/dnt003. Google Scholar [19] J. Valein and E. Zuazua, Stabilization of the wave equation on 1-D networks,, SIAM J. Control Optim., 48 (2009), 2771. doi: 10.1137/080733590. Google Scholar [20] Y. Zhang and G. Xu, Controller design for bush-type 1-D wave networks,, ESAIM Control Optim. Calc. Var., 18 (2012), 208. doi: 10.1051/cocv/2010050. Google Scholar [21] Y. Zhang and G. Xu, Exponential and super stability of a wave network,, Acta Appl. Math., 124 (2013), 19. doi: 10.1007/s10440-012-9768-1. Google Scholar [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. doi: 10.1137/060649367. Google Scholar

show all references

##### References:
 [1] K. Ammari and M. Jellouli, Stabilization of star-shaped networks of strings,, Differential and Integral Equations, 17 (2004), 1395. Google Scholar [2] K. Ammari, M. Jellouli and M. Khenissi, Stabilization of generic trees of strings,, J. Dyn. Control Syst., 11 (2005), 177. doi: 10.1007/s10883-005-4169-7. Google Scholar [3] A. Bacciotti and L. Rosier, Liapunov Functions and Stability in Control Theory,, $2^{nd}$ edition, (2005). doi: 10.1007/b139028. Google Scholar [4] S. P. Bhat and D. S. Bernstein, Finite-time stability of continuous autonomous systems,, SIAM J. Control Optim., 38 (2000), 751. doi: 10.1137/S0363012997321358. Google Scholar [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. doi: 10.1512/iumj.1995.44.2001. Google Scholar [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. doi: 10.1016/S0764-4442(01)01942-5. Google Scholar [7] R. Dáger and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-Structures,, Mathématiques & Applications (Berlin) [Mathematics & Applications], (2006). doi: 10.1007/3-540-37726-3. Google Scholar [8] M. Gugat, Optimal boundary feedback stabilization of a string with moving boundary,, IMA J. Math. Control Inform., 25 (2008), 111. doi: 10.1093/imamci/dnm014. Google Scholar [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. doi: 10.1137/100799824. Google Scholar [10] M. Gugat and M. Sigalotti, Stars of vibrating strings: Switching boundary feedback stabilization,, Netw. Heterog. Media, 5 (2010), 299. doi: 10.3934/nhm.2010.5.299. Google Scholar [11] V. T Haimo, Finite time controllers,, SIAM J. Control Optim., 24 (1986), 760. doi: 10.1137/0324047. Google Scholar [12] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method,, RAM: Research in Applied Mathematics, (1994). Google Scholar [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, (1994). doi: 10.1007/978-1-4612-0273-8. Google Scholar [14] A. Majda, Disappearing solutions for the dissipative wave equation,, Indiana Univ. Math. J., 24 (): 1119. Google Scholar [15] E. Moulay and W. Perruquetti, Finite-time stability and stabilization: state of the art,, in Advances in Variable Structure and Sliding Mode Control, (2006), 23. doi: 10.1007/11612735_2. Google Scholar [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. Google Scholar [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. doi: 10.1137/130910762. Google Scholar [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. doi: 10.1093/imamci/dnt003. Google Scholar [19] J. Valein and E. Zuazua, Stabilization of the wave equation on 1-D networks,, SIAM J. Control Optim., 48 (2009), 2771. doi: 10.1137/080733590. Google Scholar [20] Y. Zhang and G. Xu, Controller design for bush-type 1-D wave networks,, ESAIM Control Optim. Calc. Var., 18 (2012), 208. doi: 10.1051/cocv/2010050. Google Scholar [21] Y. Zhang and G. Xu, Exponential and super stability of a wave network,, Acta Appl. Math., 124 (2013), 19. doi: 10.1007/s10440-012-9768-1. Google Scholar [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. doi: 10.1137/060649367. Google Scholar
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