The approximate boundary synchronization by $p$-groups $(p\ge 1)$ in the pinning sense has been introduced in [T.-T. Li and B. Rao, Asymp. Anal. 86 (2014) 199–224], in this paper the authors give a new and more natural definition on the approximate boundary synchronization by $p$-groups in the consensus sense for a coupled system of $N$ wave equations with Dirichlet boundary controls. We show that the approximate boundary synchronization by $p$-groups in the consensus sense is equivalent to that in the pinning sense. Moreover, by means of a corresponding Kalman’s criterion, the concept of the number of total (direct and indirect) controls is introduced. It turns out that in the case that the minimal number of total controls is equal to $\left(N-p\right)$, the existence of the approximately synchronizable state by $p$-groups as well as the necessity of the strong ${C}_{p}$-compatibility condition are the consequence of the approximate boundary synchronization by $p$-groups, while, in the opposite case, the approximate boundary synchronization by $p$-groups could imply some non-expected additional properties, called the induced approximate boundary synchronization.

Keywords: Direct and indirect controls, system of wave equations, Dirichlet boundary controls, approximate boundary synchronization by groups, strong C

_{p}-compatibility condition, Kalman’s criterioninduced approximate synchronization

^{1}; Rao, Bopeng

^{1}

@article{COCV_2018__24_4_1675_0, author = {Li, Tatsien and Rao, Bopeng}, title = {On the approximate boundary synchronization for a coupled system of wave equations: direct and indirect controls}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {1675--1704}, publisher = {EDP-Sciences}, volume = {24}, number = {4}, year = {2018}, doi = {10.1051/cocv/2017043}, mrnumber = {3922429}, zbl = {1415.93022}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2017043/} }

TY - JOUR AU - Li, Tatsien AU - Rao, Bopeng TI - On the approximate boundary synchronization for a coupled system of wave equations: direct and indirect controls JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2018 SP - 1675 EP - 1704 VL - 24 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2017043/ DO - 10.1051/cocv/2017043 LA - en ID - COCV_2018__24_4_1675_0 ER -

%0 Journal Article %A Li, Tatsien %A Rao, Bopeng %T On the approximate boundary synchronization for a coupled system of wave equations: direct and indirect controls %J ESAIM: Control, Optimisation and Calculus of Variations %D 2018 %P 1675-1704 %V 24 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2017043/ %R 10.1051/cocv/2017043 %G en %F COCV_2018__24_4_1675_0

Li, Tatsien; Rao, Bopeng. On the approximate boundary synchronization for a coupled system of wave equations: direct and indirect controls. ESAIM: Control, Optimisation and Calculus of Variations, Volume 24 (2018) no. 4, pp. 1675-1704. doi : 10.1051/cocv/2017043. http://archive.numdam.org/articles/10.1051/cocv/2017043/

[1] A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems. SIAM J. Control Optimiz. 42 (2003) 871–903 | DOI | Zbl

,[2] A hierarchic multi-level energy method for the control of bidiagonal and mixed n-coupled cascade systems of PDE’s by a reduced number of controls. Adv. Diff. Equ. 18 (2013) 1005–1072 | MR | Zbl

,[3] Null-controllability of some reaction-diffusion systems with one control force. J. Math. Anal. Appl. 320 (2006) 928–44 | DOI | MR | Zbl

, and ,[4] Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J. Control Optimiz. 30 (1992) 1024–63 | DOI | MR | Zbl

, and ,[5] Controllability of two coupled wave equations on a compact manifold. Arch. Ration. Mech. Anal. 211 (2014) 113–187 | DOI | MR | Zbl

and ,[6] A systematic method for building smooth controls for smooth data. Discrete Contin. Dyn. Syst. Ser. B 14 (2010) 1375–1401 | MR | Zbl

and ,[7] Boundary controllability of parabolic coupled equations. J. Func Anal. 259 (2010) 1720–1758 | DOI | MR | Zbl

, and ,[8] Controllability and observability conditions for linear autonomous systems Indag. Math. )N.S.) 31 (1969) 443–5 | MR | Zbl

,[9] Oeuvres Complètes, Vol.15, Swets & Zeitlinger B.V., Amsterdam (1964)

,[10] Contributions to the theory of optimal control. Bol. Soc. Mat. Mexicana 5 (1960) 102–119 | MR | Zbl

,[11] Approximate controllability and asymptotic synchronization for a coupled system of wave equations with Neumann boundary controls. Contemporary Computational Mathematics – a celebration of the 80th Birthday of Ian Sloan (Edited by , , ), Springer-Verlag, 2018 | MR

, and ,[12] Asymptotic controllability for linear hyperbolic systems. Asymptotic Anal. 72 (2011) 169–185 | DOI | MR | Zbl

and ,[13] Exact synchronization for a coupled system of wave equations with Dirichlet boundary controls. Chin. Ann. Math. 34B (2013) 139–160; Partial differential equations: Theory, Control and Approximation, 295–321, Springer, Dordrecht 2013 | MR | Zbl

and ,[14] Asymptotic controllability and asymptotic synchronization for a coupled system of wave equations with Dirichlet boundary controls. Asymp. Anal. 86 (2014) 199–224 | MR | Zbl

and ,[15] A note on the exact synchronization by groups for a coupled system of wave equations. Math. Methods Appl. Sci. 38 (2015) 241–243 | DOI | MR | Zbl

and ,[16] Critères du type de Kalman pour la contrôlabilité approchée et la synchronisation approchée d’un système couplé d’équations des ondes. C. R. Math. Acad. Sci. Paris 353 (2015) 63–68 | DOI | MR | Zbl

and ,[17] Exact synchronization by groups for a coupled system of wave equations with Dirichlet boundary controls. J. Math. Pures Appl. 105 (2016) 86–101 | DOI | MR | Zbl

and ,[18] Kalman-type criteria for the approximate controllability and approximate synchronization of a coupled system of wave equations. SIAM J. Control Optimiz. 54 (2016) 49–72 | DOI | MR | Zbl

and ,[19] Une nouvelle approche pour la synchronizaation approchée d’un système couplé d’équations des ondes: contrôles directs et indirects. C.R. Math. Acad. Sci. Paris 354 (2016) 1006–1012 | DOI | MR | Zbl

and ,[20] Exact boundary controllability for a coupled system of wave equations with Neumann boundary controls to appear in Chin., Ann. Math. | MR

and ,[21] Generalized exact boundary synchronization for a coupled system of wave equations. Discrete Contin. Dyn. Syst. 34 (2014) 2893–2905 | DOI | MR | Zbl

, and ,[22] Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Vol. 1, Masson, Paris (1985) | MR

,[23] A spectral approach to the indirect boundary control of a system of weakly coupled wave equations. Discrete Contin. Dyn. Syst. 23 (2009) 399–413 | DOI | MR | Zbl

and ,[24] Averaged controllability for random evolution partial differential equations. J. Math. Pures Appl. 105 (2016) 367–414 | DOI | MR | Zbl

and ,[25] Exact controllability of a cascade system of conservative equations. C.R. Math. Acad. Sci. Paris 349 (2011) 291–295 | DOI | MR | Zbl

and ,[26] Controllability and stabilization theory for linear partial differential equations: Recent progress and open questions. SIAM Rev. 20 (1978) 639–739 | MR | Zbl

,[27] Averaged control. Automatica J. IFAC 50 (2014) 3077–3087 | DOI | MR | Zbl

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