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The Kalman condition for the boundary controllability of coupled 1-d wave equations

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  • The focus of this paper is the exact controllability of a system of $ N $ one-dimensional coupled wave equations when the control is exerted on a part of the boundary by means of one control. We give a Kalman condition (necessary and sufficient) and give a description of the attainable set. In general, this set is not optimal, but can be refined under certain conditions.

    Mathematics Subject Classification: Primary: 35Q93, 93B05; Secondary: 35L05, 15A18.

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  • [1] F. Alabau-Boussouira, A two-level energy method for indrect boundary observability and controllability of weakly coupled hyperbolic systems, SIAM J. Control Optim., 42 (2003), 871-906.  doi: 10.1137/S0363012902402608.
    [2] F. Alabau-Boussouira, Insensitizing exact controls for the scalar wave equation and exact controllability of 2-coupled cascade systems of PDE's by a single control, Math. Control Signals Systems, 26 (2014), 1-46.  doi: 10.1007/s00498-013-0112-8.
    [3] F. Alabau-Boussouira and M. Léautaud, Indirect controllability of locally coupled systems under geometric conditions, C. R. Acad. Sci. Paris, 349 (2011), 395-400.  doi: 10.1016/j.crma.2011.02.004.
    [4] F. Ammar-Kohdja, A. Benabdallah, M. González-Burgos and L. de Teresa, The Kalman condition for the boundary controllability of coupled parabolic systems. bounds on biorthogonal families to complex matrix exponentials, JMPA, 96 (2011), 555–590, https://doi.org/10.1016/j.matpur.2011.06.005. doi: 10.1016/j.matpur.2011.06.005.
    [5] S. Avdonin, A. Choque and L. de Teresa, Exact boundary controllability results for two coupled 1-d hyperbolic equations, Int. J. Appl. Math. Comput. Sci., 23 (2013), 701–710, https://doi.org/10.2478/amcs-2013-0052. doi: 10.2478/amcs-2013-0052.
    [6] S. Avdonin and L. de Teresa, The Kalman Condition for the Boundary Controllability of Coupled 1-d Wave Equations, arXiv E-Prints, arXiv: 1902.08682.
    [7] S. Avdonin and J. Edward, Exact controllability for string with attached masses, SIAM J. Control Optim., 56 (2018), 945-980.  doi: 10.1137/15M1029333.
    [8] S. A. Avdonin and  S. A. IvanovFamilies of Exponentials: The Method of Moments in Controllability Problems for Distributed Parameter Systems, Cambring University Press, 1995. 
    [9] S. A. Avdonin and S. A. Ivanov, Exponential Riesz bases of subspaces and divided differences, St. Petersburg Mathematical Journal, 13 (2002), 339-351. 
    [10] S. Avdonin and W. Moran, Ingham type inequalities and Riesz bases of subspaces and divided differences, Int. J. Appl. Math. Compt. Sci., 11 (2001), 803-820. 
    [11] A. BennourF. Ammaar Khodja and D. Tenious, Exact and approximate controllability of coupled one-dimensional hyperbolic equations, Ev. Eq. and Cont. Teho., 6 (2017), 487-516.  doi: 10.3934/eect.2017025.
    [12] H. O. Fattorini, Estimates for sequences biorthogonal to certain complex exponentials and boundary control of the wave equation, Lecture Notes in Control and Informat. Sci., 2 (1977), 111-124. 
    [13] R. E. Kalman, P. L. Palb and M. A. Arbib, Topics in Mathematical Control Theory, New York-Toronto, Ont.-London, 1969.
    [14] T. Liard and P. Lissy, A Kalman rank condition for the indirect controllability of coupled systems of linear operator groups, Math. Control Signals Syst., 29 (2017), Art. 9, 35 pp, https://doi.org/10.1007/s00498-017-0193-x. doi: 10.1007/s00498-017-0193-x.
    [15] J. Park, On the boundary controllability of coupled 1-d wave equations, Proceedings of 3rd IFAC Workshop on Control of Systems Governed by Partial Differential Equations and XI Workshop Control of Distributed Parameter Systems, Oaxaca, Mexico, May, 20–24.
    [16] L. Rosier and L. de Teresa, Exact controllability of a cascade system of conservative equations, C. R. Acad. Sci. Paris, Ser. I, 349 (2011), 291–296, https://doi.org/10.1016/j.crma.2011.01.014. doi: 10.1016/j.crma.2011.01.014.
    [17] M. Tucsnak and G. Weiss, Observation and Control of Operator Semigroups, Advanced Texts, Birkhäuser, Basel-Boston-Berlin, 2009. doi: 10.1007/978-3-7643-8994-9.
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