In this paper, we study the observability and controllability of wave equations coupled by first or zero order terms on a compact manifold. We adopt the approach in Dehman-Lebeau’s paper [B. Dehman and G. Lebeau, SIAM J. Control Optim. 48 (2009) 521–550.] to prove that: the weak observability inequality holds for wave equations coupled by first order terms on compact manifold without boundary if and only if a class of ordinary differential equations related to the symbol of the first order terms along the Hamiltonian flow are exactly controllable. We also compute the higher order part of the observability constant and the observation time. By duality, we obtain the controllability of the dual control system in a finite co-dimensional space. This gives the full controllability under the assumption of unique continuation of eigenfunctions. Moreover, these results can be applied to the systems of wave equations coupled by zero order terms of cascade structure after an appropriate change of unknowns and spaces. Finally, we provide some concrete examples as applications where the unique continuation property indeed holds.
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DOI : 10.1051/cocv/2020004
Mots-clés : Wave equation, coupled system, observability inequality, unique continuation property, controllability
@article{COCV_2020__26_1_A14_0,
author = {Cui, Yan and Laurent, Camille and Wang, Zhiqiang},
title = {On the observability inequality of coupled wave equations: the case without boundary},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {26},
year = {2020},
doi = {10.1051/cocv/2020004},
mrnumber = {4064475},
zbl = {1441.35156},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2020004/}
}
TY - JOUR AU - Cui, Yan AU - Laurent, Camille AU - Wang, Zhiqiang TI - On the observability inequality of coupled wave equations: the case without boundary JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2020 VL - 26 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2020004/ DO - 10.1051/cocv/2020004 LA - en ID - COCV_2020__26_1_A14_0 ER -
%0 Journal Article %A Cui, Yan %A Laurent, Camille %A Wang, Zhiqiang %T On the observability inequality of coupled wave equations: the case without boundary %J ESAIM: Control, Optimisation and Calculus of Variations %D 2020 %V 26 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2020004/ %R 10.1051/cocv/2020004 %G en %F COCV_2020__26_1_A14_0
Cui, Yan; Laurent, Camille; Wang, Zhiqiang. On the observability inequality of coupled wave equations: the case without boundary. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 14. doi : 10.1051/cocv/2020004. http://www.numdam.org/articles/10.1051/cocv/2020004/
[1] , Indirect boundary stabilization of weakly coupled hyperbolic systems. SIAM J. Control Optim. 41 (2002) 511–541. | DOI | MR | Zbl
[2] , A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems. SIAM J. Control Optim. 42 (2003) 871–906. | DOI | MR | Zbl
[3] , 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 Syst. 26 (2014) 1–46. | DOI | MR | Zbl
[4] and , Indirect stabilization of locally coupled wave-type systems. ESAIM Control Optim. Calc. Var. 18 (2012) 548–582. | DOI | Numdam | MR | Zbl
[5] and , Indirect controllability of locally coupled wave-type systems and applications. J. Math. Pures Appl. 99 (2013) 544–576. | DOI | MR | Zbl
[6] , and , Internal Controllability of First Order Quasi-linear Hyperbolic Systems with a Reduced Number of Controls. SIAM J. Control Optim. 55 (2017) 300–323. | DOI | MR | Zbl
[7] , and , A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities. ESAIM Control Optim. Calc. Var. 23 (2017) 721–749. | DOI | Numdam | MR | Zbl
[8] and , Stabilization of two coupled wave equations on a compact manifold with boundary. J. Math. Anal. Appl. 436 (2016) 944–969. | DOI | MR | Zbl
[9] , , and The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials. J. Math. Pures Appl. 96 (2011) 555–590. | DOI | MR | Zbl
[10] , , and , Null-controllability of some systems of parabolic type by one control force. ESAIM Control Optim. Calc. Var. 11 (2005) 426–448. | DOI | Numdam | MR | Zbl
[11] , , and , A Kalman rank condition for the localized distributed controllability of a class of linear parbolic systems. J. Evol. Equ. 9 (2009) 267–291. | DOI | MR | Zbl
[12] , , and , Recent results on the controllability of linear coupled parabolic problems: a survey. Math. Control Relat. Fields 1 (2011) 267–306. | DOI | MR | Zbl
[13] and , Simultaneous controllability in sharp time for two elastic strings. ESAIM Control Optim. Calc. Var. 6 (2001) 259–273. | DOI | Numdam | MR | Zbl
[14] , and , Un exemple d’utilisation des notions de propagation pour le contrôle et la stabilisation de problèmes hyperboliques. Rend. Sem. Mat. Univ. Politec. Torino 11–31 (1989) 1988. | MR | Zbl
[15] , and , Microlocal ideas in control and stabilization, in Control of boundaries and stabilization, edited by Clermont-Ferrand Vol. 125 of Lecture Notes in Control and Information Sciences, Springer, Berlin (1989) 14–30. | DOI | MR | Zbl
[16] , and , Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J. Control Optim. 30 (1992) 1024–1065. | DOI | MR | Zbl
[17] and , Approximate controllability conditions for some linear 1D parabolic systems with space-dependent coefficients. Math. Control Relat. Fields 4 (2014) 263–287. | DOI | MR | Zbl
[18] , A classification of linear controllable systems. Kybernetika (Prague) 6 (1970) 173–188. | MR | Zbl
[19] and , Mesures de défaut de compacité, application au système de Lamé. Ann. Sci. École Norm. Sup. (4) 34 (2001) 817–870. | DOI | Numdam | MR | Zbl
[20] and , On the control of coupled wave equations: the case with boundary, in preparation, 2018.
[21] and , Asymptotic stability of wave equations coupled by velocities. Math. Control Relat. Fields 6 (2016) 429–446. | DOI | MR | Zbl
[22] , Control and nonlinearity, in Mathematical Surveys and Monographs, Vol. 136 American Mathematical Society, Providence, RI (2007). | MR | Zbl
[23] , Insensitizing controls for the 1-D wave equation. SIAM J. Control Optim. 45 (2006) 1758–1768. | DOI | MR | Zbl
[24] and , Analysis of the HUM control operator and exact controllability for semilinear waves in uniform time. SIAM J. Control Optim. 48 (2009) 521–550. | DOI | MR | Zbl
[25] , and , Controllability of two coupled wave equations on a compact manifold. Arch. Ration. Mech. Anal. 211 (2014) 113–187. | DOI | MR | Zbl
[26] and , Indirect controllability of some linear parabolic systems of m equations with m − 1 controls involving coupling terms of zero or first order. J. Math. Pures Appl. 106 (2016) 905–934. | DOI | MR | Zbl
[27] and , Compact perturbations of controlled systems. Math. Control Relat. Fields 8 (2018) 397–410. | DOI | MR | Zbl
[28] and , Sharp observability estimates for heat equations. Arch. Ration. Mech. Anal. 202 (2011) 975–1017. | DOI | MR | Zbl
[29] , Null controllability of some systems of two parabolic equations with one control force. SIAM J. Control Optim. 46 (2007) 379–394. | DOI | MR | Zbl
[30] , The analysis of linear partial differential operators. III, Classics in Mathematics, Springer, Berlin (2007). Pseudo-differential operators, Reprint of the 1994 edition. | DOI | MR | Zbl
[31] , Best exponential decay rate of energy for the vectorial damped wave equation. SIAM J. Control Optim. 56 (2017) 3432–3453. | DOI | MR | Zbl
[32] and , Uniform observability estimates for linear waves. ESAIM Control Optim. Calc. Var. 22 (2016) 1097–1136. | DOI | Numdam | MR | Zbl
[33] Équation des ondes amorties, in Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), Vol. 19 of Mathematical Physics Studies, Springer, Berlin (1996) 73–109. | MR | Zbl
[34] and , Experimental study of the HUM control operator for linear waves. Exp. Math. 19 (2010) 93–120. | DOI | MR | Zbl
[35] and , Exact synchronization for a coupled system of wave equations with Dirichlet boundary controls. Chin. Ann. Math. Ser. B 34 (2013) 139–160. | DOI | MR | Zbl
[36] and , Exact synchronization for a coupled system of wave equations with Dirichlet boundary controls, in Partial differential equations: theory, control and approximation, Springer, Dordrecht (2014) 295–321. | MR
[37] and , A Kalman rank condition for the indirect controllability of coupled systems of linear operator groups. Math. Control Signals Syst. 29 (2017) 29:9. | DOI | MR | Zbl
[38] , Exact controllability, stabilization and perturbations for distributed systems. SIAM Rev. 30 (1988) 1–68. | DOI | MR | Zbl
[39] and , Internal observability for coupled systems of linear partial differential equations 57 (2019) 832–853. | MR
[40] , and Finite codimensional controllability for evolution equations []. | arXiv
[41] , and , Null controllability of a cascade system of Schrödinger equations. Electron. J. Differ.Equ. 12 (2016) 74. | MR | Zbl
[42] , The control transmutation method and the cost of fast controls. SIAM J. Control Optim. 45 (2006) 762–772. | DOI | MR | Zbl
[43] , Semigroups of linear operators and applications to partial differential equations, in Applied Mathematical Sciences, Vol. 44, Springer-Verlag (New York 1983). | MR | Zbl
[44] , Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions. SIAM Rev. 20 (1978) 639–739. | DOI | MR | Zbl
[45] , Locally distributed desensitizing controls for the wave equation. C. R. Math. Acad. Sci. Paris 346 (2008) 407–412. | DOI | MR | Zbl
[46] , Contrôle optimal, Mathématiques Concrètes, [Concrete Mathematics], Vuibert, Paris (2005). Théorie & applications. [Theory and applications]. | MR | Zbl
[47] , On the stability of Friedrichs’ scheme and the modified Lax-Wendroff scheme. Math. Comp. 24 (1970) 767–770. | MR | Zbl
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