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April  2019, 24(4): 1653-1675. doi: 10.3934/dcdsb.2018286

On the finite-time Bhat-Bernstein feedbacks for the strings connected by point mass

 Faculté des Sciences de Bizerte and Ecole Polytechnique de Tunisie, Laboratoire LIM d'Ingénierie Mathématique., Université de Carthage-Tunisia

* Corresponding author: Ghada Ben Belgacem

Received  February 2017 Revised  April 2018 Published  August 2018

In this article, the problem of finite-time stabilization of two strings connected by point mass is discussed. We use the so-called Riemann coordinates to convert the study system into four transport equations coupled with the dynamic of the charge. We act by Bhat-Bernstein feedbacks in various positions (two extremities, the point mass and one of boundaries, only on the point mass, ...) and we show that in some cases the nature of the stability depends sensitively on the physical parameters of the system.

Citation: Ghada Ben Belgacem, Chaker Jammazi. On the finite-time Bhat-Bernstein feedbacks for the strings connected by point mass. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1653-1675. doi: 10.3934/dcdsb.2018286
References:
 [1] F. Alabau-Boussouira, V. Perrollaz and L. Rosier, Finite-time stabilization of a network of strings, Mathematical Control and Related Fields, 5 (2015), 721-742. doi: 10.3934/mcrf.2015.5.721. [2] S. Amin, F. M. Hante, A. M. Bayen, M. Egerstedt and B. Mishra, On stability of switched linear hyperbolic conservation laws with reflecting boundaries, in International Workshop on Hybrid Systems: Computation and Control, Springer, 2008,602-605. doi: 10.1007/978-3-540-78929-1_44. [3] A. Bacciotti, Local Stabilizability of Nonlinear Control Systems, vol. 8, World Scientific Publishing Co., Inc., River Edge, NJ, 1992. [4] G.B. Belgacem and C. Jammazi, On the finite-time boundary dissipative for a class of hyperbolic systems. The networks example, IFAC-PapersOnLine, 49 (2016), 186-191. doi: 10.1016/j.ifacol.2016.07.435. [5] S.P. Bhat and D.S. Bernstein, Continuous finite-time stabilization of the translational and rotational double integrators, IEEE Trans. Automatic Control, 43 (1998), 678-682. doi: 10.1109/9.668834. [6] S.P. Bhat and D.S. Bernstein, Finite-time stability of continuous autonomous systems, SIAM J. Control and Optim, 38 (2000), 751-766. doi: 10.1137/S0363012997321358. [7] S.P. Bhat and D.S. Bernstein, Geometric homogeneity with applications to finite-time stability, Math Control Signals Systems, 17 (2005), 101-127. doi: 10.1007/s00498-005-0151-x. [8] R. Carles and C. Gallo, Finite time extinction by nonlinear damping for the schrödinger equation, Comm. Partial Differential Equations, 36 (2011), 961-975. doi: 10.1080/03605302.2010.531074. [9] C. Castro, Asymptotic analysis and control of a hybrid system composed by two vibrating strings connected by a point mass, ESAIM: Control, Optim. Calc. Var, 2 (1997), 231-280. doi: 10.1051/cocv:1997108. [10] C. Castro and E. Zuazua, Une remarque sur les séries de Fourier non-harmoniques et son application sur la contrôlabilité des cordes avec densité singuliére, C. R. Acad. Sci., 323 (1996), 365-370. [11] J.-M. Coron, On the stabilization in finite time of locally controllable systems by means of continuous time-varying feedback laws, SIAM J. Control Optim, 33 (1995), 804-833. doi: 10.1137/S0363012992240497. [12] J.-M. Coron, Control and Nonlinearity, 136, American Mathematical Soc., 2007 [13] J.-M. Coron and H.-M. Nguyen, Null controllability and finite time stabilization for the heat equations with variable coefficients in space in one dimension via backstepping approach, Archive for Rational Mechanics and Analysis, 225 (2017), 993-1023. doi: 10.1007/s00205-017-1119-y. [14] J. deHalleux, C. Prieur, J.-M. Coron, B. d'Andréa Novel and G. Bastin, Boundary feedback control in networks of open channels, Automatica, 39 (2003), 1365-1376. doi: 10.1016/S0005-1098(03)00109-2. [15] L. C. Evans, Partial Differential Equations. Graduate Studies in Mathematics, vol. 19, American Mathematical Society, 1998. [16] E. Godlewski and P. A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservations Laws, vol. 118, Springer, 1996. doi: 10.1007/978-1-4612-0713-9. [17] V. Haimo, Finite time controllers, SIAM J. Control and Optim, 24 (1986), 760-770. doi: 10.1137/0324047. [18] S. Hansen and E. Zuazua, Exact controllability and stabilization of a vibrating string with an interior point mass, SIAM J. Control Optim, 33 (1995), 1357-1391. doi: 10.1137/S0363012993248347. [19] Y. Hong and Z.-P. Jiang, Finite-time stabilization of nonlinear systems with parametric and dynamic uncertainties, IEEE Trans. Autom. Cont., 51 (2006), 1950-1956. doi: 10.1109/TAC.2006.886515. [20] Y. Hong, G. Yang, D. Cheng and S. Spurgeon, Finite time convergent control using terminal sliding mode, J. Control Theory Appl., 2 (2004), 69-74. doi: 10.1007/s11768-004-0026-6. [21] Y. Hong, Finite-time stabilization and stabilizability of a class of controllable systems, Systems and control letters, 46 (2002), 231-236. doi: 10.1016/S0167-6911(02)00119-6. [22] Y. Hong, J. Huang and Y. Xu, On an output feedback finite-time stabilization problem, IEEE Trans. Autom. Cont., 46 (2001), 305-309. doi: 10.1109/9.905699. [23] L. Hu and F. D. Meglio, Finite-time backstepping boundary stabilization of 3 × 3 hyperbolic systems, in Control Conference (ECC), 2015 European, IEEE, 2015, 67-72. [24] X. Huang, W. Lin and B. Yang, Global finite-time stabilization of a class of uncertain nonlinear systems, Automatica, 41 (2005), 881-888. doi: 10.1016/j.automatica.2004.11.036. [25] C. Jammazi, A discussion on the Hölder and robust finite-time partial stabilizability of Brockett's Integrator, ESAIM Control Optim. Calc. Var., 18 (2012), 360-382. doi: 10.1051/cocv/2010101. [26] C. Jammazi, A simple proof of finite-time stabilizability of without drift systems, in Proceeding of the 52nd IEEE Conference on Decision and Control, vol. 52, Florence-Italy, 2013, 1301-1306. [27] C. Jammazi, Continuous and discontinuous homogeneous feedbacks finite-time partial stabilizing controllable multichained systems, SIAM J. Control Optim, 52 (2014), 520-544. doi: 10.1137/110856393. [28] C. Jammazi, Some results on finite-time stabilizability: Application to triangular control systems, IMA Journal of Mathematical Control and Information, 27 (2010), 29-56. doi: 10.1093/imamci/dnp025. [29] E. B. Lee and Y. You, Stabilization of a vibrating string linked by point masses, in Control of Boundaries and Stabilization, Springer, 125 (1989), 177-198. doi: 10.1007/BFb0043360. [30] W. Littman and L. Markus, Exact boundary controllability of a hybrid system of elasticity, Archive for Rational Mechanics and Analysis, 103 (1988), 193-236. doi: 10.1007/BF00251758. [31] W. Littman and L. Markus, Stabilization of a hybrid system of elasticity by feedback boundary damping, Annali di Matematica Pura ed Applicata, 152 (1988), 281-330. doi: 10.1007/BF01766154. [32] W. Littman and S.W. Taylor, Boundary feedback stabilization of a vibrating string with an inerior point mass, Nonlinear Problems in Mathematical Physics and Related Topics Ⅰ, 1 (2002), 271-287. doi: 10.1007/978-1-4615-0777-2_16. [33] E. Moulay, Stability and stabilization of homogeneous systems depending on a parameter, IEEE Trans. Autom. Cont, 54 (2009), 1342-1385. doi: 10.1109/TAC.2009.2015560. [34] H. Mounier, J. Rudolph, M. Fliess and P. Rouchon, Tracking control of vibrating string with an interior mass viewed as delay system, ESSAIM: COCV, 3 (1998), 315-321. doi: 10.1051/cocv:1998112. [35] Y. Orlov, Finite time stability and robust control synthesis of uncertain switched systems, SIAM J. Control Optim., 43 (2005), 1253-1273. doi: 10.1137/S0363012903425593. [36] Y. Orlov, Discontinuous systems: Lyapunov Analysis and Robust Synthesis under Uncertainty Conditions, Springer, 2009. [37] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, 1st edition, Applied Mathematical Sciences 44, Springer-Verlag New York, 1983 doi: 10.1007/978-1-4612-5561-1. [38] V. Perrollaz and L. Rosier, Finite time stabilization of $2×2$ hyperbolic systems on tree-shaped networks, SIAM J. Control Optim, 52 (2014), 143-163. doi: 10.1137/130910762. [39] A. Pisano, Y. Orlov and E. Usai, Traking control of the uncertain heat and wave equation via power-fractional and sliding-mode techniques, SIAM J. Control Optim, 49 (2011), 363-382. doi: 10.1137/090781140. [40] A. Polyakov, J.-M. Coron and L. Rosier, On finite-time stabilization of evolution equations: A homogeneous approach, in IEEE 55th Conference on Decision and Control (CDC), 2016, 3143-3148. doi: 10.1109/CDC.2016.7798740. [41] A. Polyakov, D. Efimov, E. Fridman and W. Perruquetti, On homogeneous distributed parameter systems, IEEE Trans. Autom. Cont, 61 (2016), 3657-3662. doi: 10.1109/TAC.2016.2525925. [42] E. Schmidt, On the modelling and exact controllability of networks of vibrating strings and masses, SIAM J. Control Optim, 30 (1992), 229-245. doi: 10.1137/0330015. [43] E. Schmidt and W. Ming, On the Modelling and Analysis of Networks of Vibrating Strings and Masses, Department of Mathematics and Statistics, McGill University, 1991. [44] D. Serre, Systems of Conservation Laws 1: Hyperbolicity, Entropies, Shocks Waves, Cambridge University Press, 1999. doi: 10.1017/CBO9780511612374. [45] Y. Shang, D. Liu and G. Xu, Super-stability and the spectrum of one-dimensional wave equations on general feedback controlled networks, IMA Journal of Mathematical Control and Information, 31 (2013), 73-99. doi: 10.1093/imamci/dnt003. [46] C.-Z. Xu and G. Sallet, Exponential stability and transfer functions of process governed by symmetric hyperbolic systems, ESAIM: Control Optim. Calc. var, 7 (2002), 421-442. doi: 10.1051/cocv:2002062. [47] 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.

show all references

References:
 [1] F. Alabau-Boussouira, V. Perrollaz and L. Rosier, Finite-time stabilization of a network of strings, Mathematical Control and Related Fields, 5 (2015), 721-742. doi: 10.3934/mcrf.2015.5.721. [2] S. Amin, F. M. Hante, A. M. Bayen, M. Egerstedt and B. Mishra, On stability of switched linear hyperbolic conservation laws with reflecting boundaries, in International Workshop on Hybrid Systems: Computation and Control, Springer, 2008,602-605. doi: 10.1007/978-3-540-78929-1_44. [3] A. Bacciotti, Local Stabilizability of Nonlinear Control Systems, vol. 8, World Scientific Publishing Co., Inc., River Edge, NJ, 1992. [4] G.B. Belgacem and C. Jammazi, On the finite-time boundary dissipative for a class of hyperbolic systems. The networks example, IFAC-PapersOnLine, 49 (2016), 186-191. doi: 10.1016/j.ifacol.2016.07.435. [5] S.P. Bhat and D.S. Bernstein, Continuous finite-time stabilization of the translational and rotational double integrators, IEEE Trans. Automatic Control, 43 (1998), 678-682. doi: 10.1109/9.668834. [6] S.P. Bhat and D.S. Bernstein, Finite-time stability of continuous autonomous systems, SIAM J. Control and Optim, 38 (2000), 751-766. doi: 10.1137/S0363012997321358. [7] S.P. Bhat and D.S. Bernstein, Geometric homogeneity with applications to finite-time stability, Math Control Signals Systems, 17 (2005), 101-127. doi: 10.1007/s00498-005-0151-x. [8] R. Carles and C. Gallo, Finite time extinction by nonlinear damping for the schrödinger equation, Comm. Partial Differential Equations, 36 (2011), 961-975. doi: 10.1080/03605302.2010.531074. [9] C. Castro, Asymptotic analysis and control of a hybrid system composed by two vibrating strings connected by a point mass, ESAIM: Control, Optim. Calc. Var, 2 (1997), 231-280. doi: 10.1051/cocv:1997108. [10] C. Castro and E. Zuazua, Une remarque sur les séries de Fourier non-harmoniques et son application sur la contrôlabilité des cordes avec densité singuliére, C. R. Acad. Sci., 323 (1996), 365-370. [11] J.-M. Coron, On the stabilization in finite time of locally controllable systems by means of continuous time-varying feedback laws, SIAM J. Control Optim, 33 (1995), 804-833. doi: 10.1137/S0363012992240497. [12] J.-M. Coron, Control and Nonlinearity, 136, American Mathematical Soc., 2007 [13] J.-M. Coron and H.-M. Nguyen, Null controllability and finite time stabilization for the heat equations with variable coefficients in space in one dimension via backstepping approach, Archive for Rational Mechanics and Analysis, 225 (2017), 993-1023. doi: 10.1007/s00205-017-1119-y. [14] J. deHalleux, C. Prieur, J.-M. Coron, B. d'Andréa Novel and G. Bastin, Boundary feedback control in networks of open channels, Automatica, 39 (2003), 1365-1376. doi: 10.1016/S0005-1098(03)00109-2. [15] L. C. Evans, Partial Differential Equations. Graduate Studies in Mathematics, vol. 19, American Mathematical Society, 1998. [16] E. Godlewski and P. A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservations Laws, vol. 118, Springer, 1996. doi: 10.1007/978-1-4612-0713-9. [17] V. Haimo, Finite time controllers, SIAM J. Control and Optim, 24 (1986), 760-770. doi: 10.1137/0324047. [18] S. Hansen and E. Zuazua, Exact controllability and stabilization of a vibrating string with an interior point mass, SIAM J. Control Optim, 33 (1995), 1357-1391. doi: 10.1137/S0363012993248347. [19] Y. Hong and Z.-P. Jiang, Finite-time stabilization of nonlinear systems with parametric and dynamic uncertainties, IEEE Trans. Autom. Cont., 51 (2006), 1950-1956. doi: 10.1109/TAC.2006.886515. [20] Y. Hong, G. Yang, D. Cheng and S. Spurgeon, Finite time convergent control using terminal sliding mode, J. Control Theory Appl., 2 (2004), 69-74. doi: 10.1007/s11768-004-0026-6. [21] Y. Hong, Finite-time stabilization and stabilizability of a class of controllable systems, Systems and control letters, 46 (2002), 231-236. doi: 10.1016/S0167-6911(02)00119-6. [22] Y. Hong, J. Huang and Y. Xu, On an output feedback finite-time stabilization problem, IEEE Trans. Autom. Cont., 46 (2001), 305-309. doi: 10.1109/9.905699. [23] L. Hu and F. D. Meglio, Finite-time backstepping boundary stabilization of 3 × 3 hyperbolic systems, in Control Conference (ECC), 2015 European, IEEE, 2015, 67-72. [24] X. Huang, W. Lin and B. Yang, Global finite-time stabilization of a class of uncertain nonlinear systems, Automatica, 41 (2005), 881-888. doi: 10.1016/j.automatica.2004.11.036. [25] C. Jammazi, A discussion on the Hölder and robust finite-time partial stabilizability of Brockett's Integrator, ESAIM Control Optim. Calc. Var., 18 (2012), 360-382. doi: 10.1051/cocv/2010101. [26] C. Jammazi, A simple proof of finite-time stabilizability of without drift systems, in Proceeding of the 52nd IEEE Conference on Decision and Control, vol. 52, Florence-Italy, 2013, 1301-1306. [27] C. Jammazi, Continuous and discontinuous homogeneous feedbacks finite-time partial stabilizing controllable multichained systems, SIAM J. Control Optim, 52 (2014), 520-544. doi: 10.1137/110856393. [28] C. Jammazi, Some results on finite-time stabilizability: Application to triangular control systems, IMA Journal of Mathematical Control and Information, 27 (2010), 29-56. doi: 10.1093/imamci/dnp025. [29] E. B. Lee and Y. You, Stabilization of a vibrating string linked by point masses, in Control of Boundaries and Stabilization, Springer, 125 (1989), 177-198. doi: 10.1007/BFb0043360. [30] W. Littman and L. Markus, Exact boundary controllability of a hybrid system of elasticity, Archive for Rational Mechanics and Analysis, 103 (1988), 193-236. doi: 10.1007/BF00251758. [31] W. Littman and L. Markus, Stabilization of a hybrid system of elasticity by feedback boundary damping, Annali di Matematica Pura ed Applicata, 152 (1988), 281-330. doi: 10.1007/BF01766154. [32] W. Littman and S.W. Taylor, Boundary feedback stabilization of a vibrating string with an inerior point mass, Nonlinear Problems in Mathematical Physics and Related Topics Ⅰ, 1 (2002), 271-287. doi: 10.1007/978-1-4615-0777-2_16. [33] E. Moulay, Stability and stabilization of homogeneous systems depending on a parameter, IEEE Trans. Autom. Cont, 54 (2009), 1342-1385. doi: 10.1109/TAC.2009.2015560. [34] H. Mounier, J. Rudolph, M. Fliess and P. Rouchon, Tracking control of vibrating string with an interior mass viewed as delay system, ESSAIM: COCV, 3 (1998), 315-321. doi: 10.1051/cocv:1998112. [35] Y. Orlov, Finite time stability and robust control synthesis of uncertain switched systems, SIAM J. Control Optim., 43 (2005), 1253-1273. doi: 10.1137/S0363012903425593. [36] Y. Orlov, Discontinuous systems: Lyapunov Analysis and Robust Synthesis under Uncertainty Conditions, Springer, 2009. [37] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, 1st edition, Applied Mathematical Sciences 44, Springer-Verlag New York, 1983 doi: 10.1007/978-1-4612-5561-1. [38] V. Perrollaz and L. Rosier, Finite time stabilization of $2×2$ hyperbolic systems on tree-shaped networks, SIAM J. Control Optim, 52 (2014), 143-163. doi: 10.1137/130910762. [39] A. Pisano, Y. Orlov and E. Usai, Traking control of the uncertain heat and wave equation via power-fractional and sliding-mode techniques, SIAM J. Control Optim, 49 (2011), 363-382. doi: 10.1137/090781140. [40] A. Polyakov, J.-M. Coron and L. Rosier, On finite-time stabilization of evolution equations: A homogeneous approach, in IEEE 55th Conference on Decision and Control (CDC), 2016, 3143-3148. doi: 10.1109/CDC.2016.7798740. [41] A. Polyakov, D. Efimov, E. Fridman and W. Perruquetti, On homogeneous distributed parameter systems, IEEE Trans. Autom. Cont, 61 (2016), 3657-3662. doi: 10.1109/TAC.2016.2525925. [42] E. Schmidt, On the modelling and exact controllability of networks of vibrating strings and masses, SIAM J. Control Optim, 30 (1992), 229-245. doi: 10.1137/0330015. [43] E. Schmidt and W. Ming, On the Modelling and Analysis of Networks of Vibrating Strings and Masses, Department of Mathematics and Statistics, McGill University, 1991. [44] D. Serre, Systems of Conservation Laws 1: Hyperbolicity, Entropies, Shocks Waves, Cambridge University Press, 1999. doi: 10.1017/CBO9780511612374. [45] Y. Shang, D. Liu and G. Xu, Super-stability and the spectrum of one-dimensional wave equations on general feedback controlled networks, IMA Journal of Mathematical Control and Information, 31 (2013), 73-99. doi: 10.1093/imamci/dnt003. [46] C.-Z. Xu and G. Sallet, Exponential stability and transfer functions of process governed by symmetric hyperbolic systems, ESAIM: Control Optim. Calc. var, 7 (2002), 421-442. doi: 10.1051/cocv:2002062. [47] 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.
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