Indirect stabilization of locally coupled wave-type systems
ESAIM: Control, Optimisation and Calculus of Variations, Volume 18 (2012) no. 2, p. 548-582

We study in an abstract setting the indirect stabilization of systems of two wave-like equations coupled by a localized zero order term. Only one of the two equations is directly damped. The main novelty in this paper is that the coupling operator is not assumed to be coercive in the underlying space. We show that the energy of smooth solutions of these systems decays polynomially at infinity, whereas it is known that exponential stability does not hold (see [F. Alabau, P. Cannarsa and V. Komornik, J. Evol. Equ. 2 (2002) 127-150]). We give applications of our result to locally or boundary damped wave or plate systems. In any space dimension, we prove polynomial stability under geometric conditions on both the coupling and the damping regions. In one space dimension, the result holds for arbitrary non-empty open damping and coupling regions, and in particular when these two regions have an empty intersection. Hence, indirect polynomial stability holds even though the feedback is active in a region in which the coupling vanishes and vice versa.

DOI : https://doi.org/10.1051/cocv/2011106
Classification:  35L05,  35L51,  93C20,  93D20
Keywords: stabilization, indirect damping, hyperbolic systems, wave equation
@article{COCV_2012__18_2_548_0,
     author = {Alabau-Boussouira, Fatiha and L\'eautaud, Matthieu},
     title = {Indirect stabilization of locally coupled wave-type systems},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {18},
     number = {2},
     year = {2012},
     pages = {548-582},
     doi = {10.1051/cocv/2011106},
     zbl = {1259.35034},
     mrnumber = {2954638},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2012__18_2_548_0}
}
Alabau-Boussouira, Fatiha; Léautaud, Matthieu. Indirect stabilization of locally coupled wave-type systems. ESAIM: Control, Optimisation and Calculus of Variations, Volume 18 (2012) no. 2, pp. 548-582. doi : 10.1051/cocv/2011106. http://www.numdam.org/item/COCV_2012__18_2_548_0/

[1] F. Alabau, P. Cannarsa and V. Komornik, Indirect internal damping of coupled systems. J. Evol. Equ. 2 (2002) 127-150. | MR 1914654 | Zbl 1011.35018

[2] F. Alabau-Boussouira, Stabilisation frontière indirecte de systèmes faiblement couplés. C. R. Acad. Sci. Paris Sér. I Math. 328 (1999) 1015-1020. | MR 1696198 | Zbl 0934.93056

[3] F. Alabau-Boussouira, Indirect boundary stabilization of weakly coupled systems. SIAM J. Control Optim. 41 (2002) 511-541. | MR 1920269 | Zbl 1031.35023

[4] F. Alabau-Boussouira, A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems. SIAM J. Control Optim. 42 (2003) 871-906. | MR 2002139 | Zbl 1125.93311

[5] F. Alabau-Boussouira, Une formule générale pour le taux de décroissance des systèmes dissipatifs non linéaires. C. R. Acad. Sci. Paris Sér. I Math. 338 (2004) 35-40. | MR 2038081 | Zbl 1073.35164

[6] F. Alabau-Boussouira, Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems. Appl. Math. Optim. 51 (2005) 61-105. | MR 2101382 | Zbl 1107.35077

[7] F. Alabau-Boussouira, Piecewise multiplier method and nonlinear integral inequalities for Petrowsky equation with nonlinear dissipation. J. Evol. Equ. 6 (2006) 95-112. | MR 2215634 | Zbl 1117.34054

[8] F. Alabau-Boussouira, Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control. NoDEA 14 (2007) 643-669. | MR 2374204 | Zbl 1147.35055

[9] F. Ammar-Khodja, A. Benabdallah, J.E. Muñoz Rivera and R. Racke, Energy decay for Timoshenko systems of memory type. J. Differ. Equ. 194 (2003) 82-115. | MR 2001030 | Zbl 1131.74303

[10] F. Ammar-Khodja, A. Benabdallah and C. Dupaix, Null controllability of some reaction-diffusion systems with one control force. J. Math. Anal. Appl. 320 (2006) 928-943. | MR 2226005 | Zbl 1157.93004

[11] 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. | MR 1178650 | Zbl 0786.93009

[12] A. Beyrath, Indirect linear locally distributed damping of coupled systems. Bol. Soc. Parana. Math. 22 (2004) 17-34. | MR 2190130 | Zbl 1091.35514

[13] N. Burq, Décroissance de l'énergie locale de l'équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel. Acta Math. 180 (1998) 1-29. | MR 1618254 | Zbl 0918.35081

[14] N. Burq and G. Lebeau, Mesures de défaut de compacité, application au système de Lamé. Ann. Sci. Éc. Norm. Supér. 34 (2001) 817-870. | Numdam | MR 1872422 | Zbl 1043.35009

[15] F. Conrad and B. Rao, Decay of solutions of the wave equation in a star-shaped domain with nonlinear boundary feedback. Asymptotic Anal. 7 (1993) 159-177. | MR 1226972 | Zbl 0791.35011

[16] L. De Teresa, Insensitizing controls for a semilinear heat equation. Commun. Part. Differ. Equ. 25 (2000) 39-72. | MR 1737542 | Zbl 0942.35028

[17] M. González-Burgos and R. Pérez-García, Controllability results for some nonlinear coupled parabolic systems by one control force. Asymptotic Anal. 46 (2006) 123-162. | MR 2205238 | Zbl 1124.35026

[18] A. Haraux, Semi-groupes linéaires et équations d'évolution linéaires périodiques. Publication du Laboratoire d'Analyse Numérique, Université Pierre et Marie Curie 78011 (1978) 123-162. | Numdam | Zbl 0341.35071

[19] O. Kavian and L. De Teresa, Unique continuation principle for systems of parabolic equations. ESAIM : COCV 16 (2010) 247-274. | Numdam | MR 2654193 | Zbl 1195.35080

[20] V. Komornik, Exact controllability and stabilization : The multiplier method. Research in Applied Mathematics 36, Masson, Paris (1994). | MR 1359765 | Zbl 0937.93003

[21] J.E. Lagnese, Boundary stabilization of thin plates, SIAM Studies in Applied Mathematics 10. SIAM (1989). | MR 1061153 | Zbl 0696.73034

[22] M. Léautaud, Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems. J. Funct. Anal. 258 (2010) 2739-2778. | MR 2593342 | Zbl 1185.35153

[23] G. Lebeau, Équation des ondes amorties, in Algebraic and geometric methods in mathematical physics. Math. Phys. Stud., Kluwer Acad. Publ., Dordrecht (1996) 73-109. | MR 1385677 | Zbl 0863.58068

[24] G. Lebeau and L. Robbiano, Stabilisation de l'équation des ondes par le bord. Duke Math. J. 86 (1997) 465-491. | MR 1432305 | Zbl 0884.58093

[25] G. Lebeau and E. Zuazua, Decay rates for the three-dimensional linear system of thermoelasticity. Arch. Rational Mech. Anal. 148 (1999) 179-231. | MR 1716306 | Zbl 0939.74016

[26] J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués, Recherches en Mathématiques Appliquées, Tome 1, 8. Masson, Paris (1988). | MR 963060 | Zbl 0653.93002

[27] K. Liu, Locally distributed control and damping for the conservative systems. SIAM J. Control Optim. 35 (1997) 1574-1590. | MR 1466917 | Zbl 0891.93016

[28] P. Martinez, A new method to obtain decay rate estimates for dissipative systems with localized damping. Rev. Math. Comput. 12 (1999) 251-283. | MR 1698906 | Zbl 0940.35034

[29] L. Miller, Escape function conditions for the observation, control, and stabilization of the wave equation. SIAM J. Control Optim. 41 (2002) 1554-1566. | MR 1971962 | Zbl 1032.35117

[30] W. Youssef, Contrôle et stabilisation de systèmes élastiques couplés. Ph. D. thesis, University of Metz, France (2009).

[31] E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping. Commun. Part. Differ. Equ. 15 (1990) 205-235. | MR 1032629 | Zbl 0716.35010