Abstract
In this paper, the exact synchronization for a coupled system of wave equations with Dirichlet boundary controls and some related concepts are introduced. By means of the exact null controllability of a reduced coupled system, under certain conditions of compatibility, the exact synchronization, the exact synchronization by groups, and the exact null controllability and synchronization by groups are all realized by suitable boundary controls.
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References
Alabau-Boussouira, F., A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems, SIAM J. Control Optim., 42, 2003, 871–906.
Fujisaka, H. and Yamada, T., Stability theory of synchronized motion in coupled-oscillator systems, Progress of Theoretical Physics, 69, 1983, 32–47.
Garofalo, N. and Lin, F., Unique continuation for elliptic operators: a geometric-variational approach, Comm. Pure Appl. Math., 40, 1987, 347–366.
Huygens, C., Horologium Oscillatorium Sive de Motu Pendulorum ad Horologia Aptato Demonstrationes Geometricae, Apud F. Muguet, Parisiis, 1673.
Komornik, V. and Loreti, P., Fourier Series in Control Theory, Springer-Verlag, New York, 2005.
Krabs, W., On Moment Theory and Controllability of One-Dimensional Vibrating Systems and Heating Processes, Lecture Notes in Control and Information Sciences, 173, Springer-Verlag, Berlin, 1992.
Yu, L., Exact boundary controllability for a kind of second-order quasilinear hyperbolic systems and its applications, Math. Meth. Appl. Sci., 33, 2010, 273–286.
Li, T. T. and Rao, B. P., Strong (weak) exact controllability and strong (weak) exact observability for quasilinear hyperbolic systems, Chin. Ann. Math., 31B(5), 2010, 723–742.
Li, T. T. and Rao, B. P., Asymptotic controllability for linear hyperbolic systems, Asymptotic Analysis, 72, 2011, 169–187.
Lions, J. L., Controlabilité Exacte, Perturbations et Stabilisation de Systèms Distribués, Vol. 1, Masson, Paris, 1988.
Liu, Z. and Rao, B. P., A spectral approach to the indirect boundary control of a system of weakly coupled wave equations, Discrete Contin. Dyn. Syst., 23, 2009, 399–414.
Loreti, P. and Rao, B. P., Optimal energy decay rate for partially damped systems by spectral compensation, SIAM J. Control Optim., 45, 2006, 1612–1632
Mehrenberger, M., Observability of coupled systems, Acta Math. Hungar., 103, 2004, 321–348.
Wang, K., Exact boundary controllability for a kind of second-order quasilinear hyperbolic systems, Chin. Ann. Math., 32B(6), 2011, 803–822.
Young, R., An Introduction to Nonharmonic Fourier Series, Academic Press, New York, London, 1980.
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In honor of the scientific heritage of Jacques-Louis Lions
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Li, T., Rao, B. Exact Synchronization for a Coupled System of Wave Equations with Dirichlet Boundary Controls. Chin. Ann. Math. Ser. B 34, 139–160 (2013). https://doi.org/10.1007/s11401-012-0754-8
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DOI: https://doi.org/10.1007/s11401-012-0754-8