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  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 & Control Theory, 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar
[9]

S. A. Avdonin and S. A. Ivanov, Exponential Riesz bases of subspaces and divided differences, St. Petersburg Mathematical Journal, 13 (2002), 339-351. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

R. E. Kalman, P. L. Palb and M. A. Arbib, Topics in Mathematical Control Theory, New York-Toronto, Ont.-London, 1969. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar
[9]

S. A. Avdonin and S. A. Ivanov, Exponential Riesz bases of subspaces and divided differences, St. Petersburg Mathematical Journal, 13 (2002), 339-351. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

R. E. Kalman, P. L. Palb and M. A. Arbib, Topics in Mathematical Control Theory, New York-Toronto, Ont.-London, 1969. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

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