STABILIZATION OF TWO STRONGLY COUPLED HYPERBOLIC EQUATIONS IN EXTERIOR DOMAINS

[1] F. Alabau, C. Piermarco and K. Vilmos: Indirect internal stabilization of weakly coupled evolution equations, J. Evol. Equ. 2 (2002), 127–150.

[2] F. Alabau and M. Leautaud: ´ Indirect stabilization of locally coupled wave-type systems, ESAIM Control Optim. Calc. Var. 18 (2012), 548–582.

[3] F. Ammar-Khodja and A. Bader: Stabilizability of systems of one-dimensional wave equations by internal or boundary control force, SIAM J. Control Optim. 39 (2001), 1833–1851.

[4] L. Aloui and M. Daoulatli: Stabilization of two coupled wave equations on a compact manifold with boundary, J. Math. Anal. Appl. 436 (2016), 944–969.

[5] L. Aloui, S. Ibrahim and K. Nakanishi: Exponential energy decay for damped Klein-Gordon equation with nonlinearities of arbitrary growth, Comm. Partial Differential Equations 36 (2011), 797–818.

[6] L. Aloui, S. Ibrahim and M. Khenissi: Energy decay for linear dissipative wave equations in exterior domains, J. Differential Equations 259 (2015), 2061–2079.

[7] C. Bardos, G. Lebeau and J. Rauch: Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim. 30 (1992), 1024–1065.

[8] N. Burq and P. Gerard: ´ Condition n´ecessaire et suffisante pour la contrˆolabilit´exacte des ondes, C. R. Acad. Sci. Paris Ser. I Math. ´ 325 (1997), 749–752.

[9] B. Dehman, G. Lebeau and E. Zuazua: Stabilization and control for the subcritical semilinear wave equation, Ann. Sci. Ecole Norm. Sup. ´ 36 (2003), 525–551.

[10] M. Daoulatli: Behaviors of energy of solutions of two coupled wave equations with nonlinear damping on a compact manifold with boundary, arXiv:1703.00172v1.

[11] M. Daoulatli: Energy decay rates for solutions of the wave equation with linear damping in exterior domain, arXiv:1203.6780v4.

[12] P. Gerard: ´ Microlocal defect measures, Comm. Partial Differential Equations 16 (1991), 1761–1794.

[13] B. Kapitonov: Uniform stabilization and exact controllability for a class of coupled hyperbolic systems, Mat. Apl. Comput. 15 (1996), 199–212.

[14] C. Laurent and J. Romain: Stabilization for the semilinear wave equation with geometric control condition, Anal. PDE 6 (2013), 1089–1119.

[15] G. Lebeau: Equations des ondes amorties; in Algebraic and geometric Methods in Mathematical Physics, Kluwer Academic, Dordrecht, 1996, 73–109.

[16] J.L. Lions: Controlabilite exacte, perturbations et stabilisation de systemes distribues. Tome 1, Recherches en Mathematique appliquees 8, Masson, Paris, 1988.

[17] R.B. Melrose and J. Sjostrand: ¨ Singularities of boundary value problems. I, Comm. Pure Appl. Math. 31 (1978), 593–617 .

[18] R.B. Melrose and J. Sjostrand: ¨ Singularities of boundary value problems. II, Comm. Pure Appl. Math. 35 (1982), 129–168.

[19] A. Matsumura: On the asymptotic behavior of solutions of semi-linear wave equations, Publ. Res. Inst. Math. Sci. 12 (1976), 169–189.

[20] M. Nakao: Energy decay for the linear and semilinear wave equations in exterior domains with some localized dissipations, Math.Z. 238 (2001), 781–797.

[21] D. Tataru: The Xs,θ spaces and unique continuation for solutions to the semilinear wave equations, Comm. Parial Differential Equations 21 (1996), 841–887.

[22] L. Toufayli: Stabilisation polynomiale et contrˆolabilit´e exacte des ´equations des ondes par des contrˆoles indirects et dynamiques, Theese universit ` e de Strasbourg, 2013, https : ´ //tel.archives-ouvertes.fr/tel-00780215.

[23] E. Zuazua: Exponential decay for the semilinear wave equation with localized damping in unbounded domains, J.Math. Pures Appl. 70 (1992), 513–529.