# American Institute of Mathematical Sciences

March  2020, 9(1): 255-273. doi: 10.3934/eect.2020005

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

 1 Dept. of Mathematics and Statistics, University of Alaska Fairbanks, Fairbanks, AK 99775, USA 2 Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, G. U. 04510 D.F., México

Received  December 2018 Revised  March 2019 Published  March 2020 Early access  August 2019

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.

Citation: Sergei Avdonin, Jeff Park, Luz de Teresa. The Kalman condition for the boundary controllability of coupled 1-d wave equations. Evolution Equations and Control Theory, 2020, 9 (1) : 255-273. doi: 10.3934/eect.2020005
##### References:
 [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. Ivanov, Families 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. Bennour, F. 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.

show all references

##### References:
 [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. Ivanov, Families 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. Bennour, F. 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.
 [1] Hamid Maarouf. Local Kalman rank condition for linear time varying systems. Mathematical Control and Related Fields, 2022, 12 (2) : 433-446. doi: 10.3934/mcrf.2021029 [2] Takeshi Taniguchi. Exponential boundary stabilization for nonlinear wave equations with localized damping and nonlinear boundary condition. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1571-1585. doi: 10.3934/cpaa.2017075 [3] Larissa V. Fardigola. Transformation operators in controllability problems for the wave equations with variable coefficients on a half-axis controlled by the Dirichlet boundary condition. Mathematical Control and Related Fields, 2015, 5 (1) : 31-53. doi: 10.3934/mcrf.2015.5.31 [4] Umberto De Maio, Akamabadath K. Nandakumaran, Carmen Perugia. Exact internal controllability for the wave equation in a domain with oscillating boundary with Neumann boundary condition. Evolution Equations and Control Theory, 2015, 4 (3) : 325-346. doi: 10.3934/eect.2015.4.325 [5] Tatsien Li (Daqian Li). Global exact boundary controllability for first order quasilinear hyperbolic systems. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1419-1432. doi: 10.3934/dcdsb.2010.14.1419 [6] Tatsien Li, Bopeng Rao, Zhiqiang Wang. A note on the one-side exact boundary controllability for quasilinear hyperbolic systems. Communications on Pure and Applied Analysis, 2009, 8 (1) : 405-418. doi: 10.3934/cpaa.2009.8.405 [7] Peter Šepitka. Riccati equations for linear Hamiltonian systems without controllability condition. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 1685-1730. doi: 10.3934/dcds.2019074 [8] Ali Wehbe, Marwa Koumaiha, Layla Toufaily. Boundary observability and exact controllability of strongly coupled wave equations. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1269-1305. doi: 10.3934/dcdss.2021091 [9] Tatsien Li, Bopeng Rao, Zhiqiang Wang. Exact boundary controllability and observability for first order quasilinear hyperbolic systems with a kind of nonlocal boundary conditions. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 243-257. doi: 10.3934/dcds.2010.28.243 [10] Feng Qi, Bai-Ni Guo. Completely monotonic functions involving divided differences of the di- and tri-gamma functions and some applications. Communications on Pure and Applied Analysis, 2009, 8 (6) : 1975-1989. doi: 10.3934/cpaa.2009.8.1975 [11] Xiaoqiang Dai, Chao Yang, Shaobin Huang, Tao Yu, Yuanran Zhu. Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems. Electronic Research Archive, 2020, 28 (1) : 91-102. doi: 10.3934/era.2020006 [12] Alberto Bressan, Marta Lewicka. A uniqueness condition for hyperbolic systems of conservation laws. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 673-682. doi: 10.3934/dcds.2000.6.673 [13] Tatsien Li, Zhiqiang Wang. A note on the exact controllability for nonautonomous hyperbolic systems. Communications on Pure and Applied Analysis, 2007, 6 (1) : 229-235. doi: 10.3934/cpaa.2007.6.229 [14] Ning-An Lai, Jinglei Zhao. Potential well and exact boundary controllability for radial semilinear wave equations on Schwarzschild spacetime. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1317-1325. doi: 10.3934/cpaa.2014.13.1317 [15] Larissa V. Fardigola. Controllability problems for the 1-d wave equations on a half-axis with Neumann boundary control. Mathematical Control and Related Fields, 2013, 3 (2) : 161-183. doi: 10.3934/mcrf.2013.3.161 [16] Bopeng Rao, Laila Toufayli, Ali Wehbe. Stability and controllability of a wave equation with dynamical boundary control. Mathematical Control and Related Fields, 2015, 5 (2) : 305-320. doi: 10.3934/mcrf.2015.5.305 [17] Yaru Xie, Genqi Xu. The exponential decay rate of generic tree of 1-d wave equations with boundary feedback controls. Networks and Heterogeneous Media, 2016, 11 (3) : 527-543. doi: 10.3934/nhm.2016008 [18] Nguyen Thanh Long, Hoang Hai Ha, Le Thi Phuong Ngoc, Nguyen Anh Triet. Existence, blow-up and exponential decay estimates for a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions. Communications on Pure and Applied Analysis, 2020, 19 (1) : 455-492. doi: 10.3934/cpaa.2020023 [19] Kunio Hidano, Kazuyoshi Yokoyama. Global existence and blow up for systems of nonlinear wave equations related to the weak null condition. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022058 [20] Ait Ben Hassi El Mustapha, Fadili Mohamed, Maniar Lahcen. On Algebraic condition for null controllability of some coupled degenerate systems. Mathematical Control and Related Fields, 2019, 9 (1) : 77-95. doi: 10.3934/mcrf.2019004

2020 Impact Factor: 1.081