Exact Boundary Synchronization by Groups for a Kind of System of Wave Equations Coupled with Velocities

Abstract

This paper deals with the exact boundary controllability and the exact boundary synchronization for a 1-D system of wave equations coupled with velocities. These problems can not be solved directly by the usual HUM method for wave equations, however, by transforming the system into a first order hyperbolic system, the HUM method for 1-D first order hyperbolic systems, established by Li-Lu (2022) and Lu-Li (2022), can be applied to get the corresponding results.

Keywords

Exact boundary controllability / Exact boundary synchronization / Coupled system of wave equations / HUM method

References

[1] Aguilar L, Orlov Y, Pisano A. Leader-follower synchronization and ISS analysis for a network of boundary-controlled wave PDEs. IEEE Control Syst. Lett., 2021, 5: 683-688

[2] Chen Y, Zuo Z, Wang Y. Bipartite consensus for a network of wave equations with time-varying disturbances. Syst. Control Lett., 2020, 136: 1-11

[3] Demetriou M, Fahroo F. Optimisation and adaptation of synchronisation controllers for networked second-order infinite-dimensional systems. Int. J. Control, 2019, 92: 112-131

[4] Hu L, Li T T, Rao B P. Exact boundary synchronization for a coupled system of 1-D wave equations with coupled boundary conditions of dissipative type. Commun. Pure Appl. Anal., 2014, 13: 881-901

[5] Huygens C. Oeuvres Complètes, 15, 1967, Amsterdam: Swets & Zeitlinger

[6] Li T T, Lu X. Exact boundary synchronization for a kind of first order hyperbolic system. ESAIM Control Optim. Calc. Var., 2022, 28: 34

[7] Li T T, Lu X, Rao B P. Exact boundary synchronization for a coupled system of wave equations with Neumann boundary controls. Chin. Ann. Math. Ser. B, 2018, 39(2): 233-252

[8] Li, T. T., Lu, X. and Rao, B. P., Approximate Boundary Null Controllability and Approximate Boundary Synchronization for a Coupled System of Wave Equations with Neumann Boundary Controls, Contemporary Computational Mathematics — a Celebration of the 80th Birthday of Ian Sloan, Dick, J., Kuo, F. Y., Woźniakowski, H. (eds.), 2, Springer-Verlag, 2018, 837–868.

[9] Li T T, Lu X, Rao B P. Exact boundary controllability and exact boundary synchronization for a coupled system of wave equations with coupled Robin boundary controls. ESAIM Control Optim. Calc. Var., 2021, 27: S7

[10] Li T T, Rao B P. Synchronisation exacte d’un système couplé d’équations des ondes par des contrôles frontières de Dirichlet. C. R. Math. Acad. Sci. Paris, 2012, 350(15–16): 767-772

[11] Li T T, Rao B P. Exact synchronization for a coupled system of wave equation with Dirichlet boundary controls. Chin. Ann. Math. Ser. B, 2013, 34(1): 139-160

[12] Li T T, Rao B P. Asymptotic controllability and asymptotic synchronization for a coupled system of wave equations with Dirichlet boundary controls. Asymptot. Anal., 2014, 86: 199-226

[13] Li T T, Rao B P. A note on the exact synchronization by groups for a coupled system of wave equations. Math. Methods Appl. Sci., 2015, 38(2): 241-246

[14] Li T T, Rao B P. Exact synchronization by groups for a coupled system of wave equations with Dirichlet boundary control. J. Math. Pures Appl., 2016, 105: 86-101

[15] Li T T, Rao B P. On the approximate boundary synchronization for a coupled system of wave equations: Direct and indirect boundary controls. ESAIM Control Optim. Calc. Var., 2019, 24: 1675-1704

[16] Li, T. T. and Rao, B. P., Boundary Synchronization for Hyperbolic Systems, Progress in Nonlinear Differential Equations and Their Applications, Subseries in Control, 94, Birkhäuser, 2019.

[17] Li T T, Rao B P. Approximate boundary synchronization by groups for a couples system of wave equations with coupled Robin boundary conditions. ESAIM Control Optim. Calc. Var., 2021, 27: 10

[18] Li T T, Rao B P, Hu L. Exact boundary synchronization for a coupled system of 1-D wave equations. ESAIM Control Optim. Calc. Var., 2014, 20: 339-361

[19] Li T T, Rao B P, Wei Y M. Generalized exact boundary synchronization for a coupled system of wave equations. Discrete Contin. Dyn. Syst., 2014, 34: 2893-2905

[20] Lu X, Li T T. Exact boundary controllability of weak solutions for a kind of first order hyperbolic system — The constructive method. Chin. Ann. Math. Ser. B, 2021, 42(5): 643-676

[21] Lu X, Li T T. Exact boundary controllability of weak solutions for a kind of first order hyperbolic system — The HUM method. Chin. Ann. Math. Ser. B, 2022, 43(1): 1-16

[22] Lu X, Li T T, Rao B P. Exact boundary synchronization by groups for a coupled system of wave equations with coupled Robin boundary controls on a general bounded domain. SIAM J. Control Optim., 2021, 59(6): 4457-4480

[23] Wiener N. Cybernetics, or Control and Communication in the Animal and the Machine, 1961 2nd ed New York, London: The M. I. T. Press, Cambridge, Mass., John Wiley & Sons, Inc.