A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities
Alabau-Boussouira, Fatiha1 ; Wang, Zhiqiang2 ; Yu, Lixin3
1 IECL, Université de Lorraine and CNRS (UMR 7502), Délégation CNRS at LJLL UMR 7598, 57045 Metz, France.
2 School of Mathematical Sciences and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai 200433, P.R. China.
3 School of Mathematics and Information Sciences, Yantai University, Yantai 264005, P.R. China.
ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 2, pp. 721-749.

In this paper, we consider the energy decay of a damped hyperbolic system of wave-wave type which is coupled through the velocities. We are interested in the asymptotic properties of the solutions of this system in the case of indirect nonlinear damping, i.e. when only one equation is directly damped by a nonlinear damping. We prove that the total energy of the whole system decays as fast as the damped single equation. Moreover, we give a one-step general explicit decay formula for arbitrary nonlinearity. Our results shows that the damping properties are fully transferred from the damped equation to the undamped one by the coupling in velocities, different from the case of couplings through displacements as shown in [F. Alabau, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999) 1015–1020; F. Alabau, P. Cannarsa and V. Komornik, J. Evol. Equ. 2 (2002) 127–150; F. Alabau, SIAM J. Control Optim. 41 (2002) 511–541; F. Alabau-Boussouira and M. Léautaud, ESAIM: COCV 18 (2012) 548–582] for the linear damping case, and in [F. Alabau-Boussouira, NoDEA 14 (2007) 643–669] for the nonlinear damping case. The proofs of our results are based on multiplier techniques, weighted nonlinear integral inequalities and the optimal-weight convexity method of [F. Alabau-Boussouira, Appl. Math. Optim. 51 (2005) 61–105; F. Alabau-Boussouira, J. Differ. Equ. 248 (2010) 1473–1517].

Reçu le :
Accepté le :
DOI : 10.1051/cocv/2016011
Classification : 35L05, 35Lxx, 93D15, 93D20
Mots-clés : Energy decay, nonlinear damping, wave equation, plate equation, weighted nonlinear integral inequality, optimal-weight convexity method
Alabau-Boussouira, Fatiha 1 ; Wang, Zhiqiang 2 ; Yu, Lixin 3

1 IECL, Université de Lorraine and CNRS (UMR 7502), Délégation CNRS at LJLL UMR 7598, 57045 Metz, France.
2 School of Mathematical Sciences and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai 200433, P.R. China.
3 School of Mathematics and Information Sciences, Yantai University, Yantai 264005, P.R. China. 
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     title = {A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {721--749},
     publisher = {EDP-Sciences},
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Alabau-Boussouira, Fatiha; Wang, Zhiqiang; Yu, Lixin. A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities. ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 2, pp. 721-749. doi : 10.1051/cocv/2016011. http://archive.numdam.org/articles/10.1051/cocv/2016011/

F. Alabau, Stabilisation frontière indirecte de systèmes faiblement couplés. C. R. Acad. Sci. Paris Sér. I Math. 328 (1999) 1015–1020. | DOI | MR | Zbl

F. Alabau, Indirect boundary stabilization of weakly coupled systems. SIAM J. Control Optim. 41 (2002) 511–541. | DOI | MR | Zbl

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 | Zbl

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

F. Alabau-Boussouira, Piecewise multiplier method and nonlinear integral inequality for Petrowsky equation with nonlinear dissipation. J. Evol. Equ. 6 (2006) 95–112. | DOI | MR | Zbl

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

F. Alabau-Boussouira, A unified approach via convexity for optimal energy decay rates of finite and infinite dimensional vibrating damped systems with applications to semi-discretized vibrating damped systems. J. Differ. Equ. 248 (2010) 1473–1517. | DOI | MR | Zbl

F. Alabau-Boussouira, New trends towards lower energy estimates and optimality for nonlinearly damped vibrating systems. J. Differ. Equ. 249 (2010) 1145–1178. | DOI | MR | Zbl

F. Alabau-Boussouira, Strong lower energy estimates for nonlinearly damped Timoshenko beams and Petrowsky equations. NoDEA 18 (2011) 571–597. | DOI | MR | Zbl

F. Alabau-Boussouira, On some recent advances on stabilization for hyperbolic equations. Vol. 2048 of Lect. Note Math. CIME Foundation Subseries Control of Partial Differential Equations. Springer Verlag (2012) 101. | MR

F. Alabau-Boussouira and K. Ammari, Sharp energy estimates for nonlinearly locally damped PDEs via observability for the associated undamped system. J. Funct. Anal. 260 (2011) 2424–2450. | DOI | MR | Zbl

F. Alabau-Boussouira and M. Léautaud, Indirect stabilization of locally coupled wave-type systems. ESAIM: COCV 18 (2012) 548–582. | Numdam | MR | Zbl

F. Alabau-Boussouira and M. Léautaud, Indirect controllability of locally coupled wave-type systems and applications. J. Math. Pures Appl. 99 (2013) 544–576. | DOI | MR | Zbl

F. Alabau, P. Cannarsa and V. Komornik, Indirect internal damping of coupled systems. J. Evol. Equ. 2 (2002) 127–150. | MR | Zbl

F. Alabau-Boussouira, P. Cannarsa and R. Guglielmi, Indirect stabilization of weakly coupled systems with hybrid boundary conditions. Math. Control Relat. Fields 1 (2011) 413–436. | DOI | MR | Zbl

F. Alabau-Boussouira, Y. Privat and E. Trélat, Nonlinear damped partial differential equations and their uniform discretizations. Preprint (2015). | arXiv | MR

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. | DOI | MR | Zbl

M. Bellassoued, Rate of decay of solution of the wave equation with arbitrary localized nonlinear damping. J. Differ. Equ. 211 (2005) 303–332. | DOI | MR | Zbl

G. Chen, A note on boundary stabilization of the wave equation. SIAM J. Control Optim. 19 (1981) 106–113. | DOI | MR | Zbl

Y. Cui and Z. Wang, Asymptotic stability of wave equations coupled by velocities. Math. Control Relat. Fields 6 (2016) 429–446. | DOI | MR | Zbl

B. Dehman, G. Lebeau and E. Zuazua, Stabilization and control for the subcritical semilinear wave equation. Annales Scientifiques de l’École Normale Supérieure 36 (2003) 525–551. | DOI | Numdam | MR | Zbl

X. Fu, Longtime behavior of the hyperbolic equations with an arbitrary internal damping. Z. Angew. Math. Phys. 62 (2011) 667–680. | DOI | MR | Zbl

X. Fu, Sharp decay rates for the weakly coupled hyperbolic system with one internal damping. SIAM J. Control Optim. 50 (2012) 1643–1660. | DOI | MR | Zbl

R. Joly and C. Laurent, Stabilization for the semilinear wave equation with geometric control condition. Ann. PDE 6 (2012) 1089–1119. | DOI | MR | Zbl

J.U. Kim and Y. Renardy, Boundary control of the Timoshenko beam. SIAM J. Control Optim. 25 (1987) 1417–1429. | DOI | MR | Zbl

V. Komornik, Exact controllability and stabilization: The Multiplier Method. Vol. 36 of Collection RMA. Masson-John Wiley, Paris-Chicester (1994). | MR | Zbl

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping. Differ. Integral Equ. 8 (1993) 507–533. | MR | Zbl

G. Lebeau, Équation des ondes amorties, Algebraic and geometric methods in mathematical physics (Kaciveli, 1993). Math. Phys. Study. Kluwer Acad. Publ., Dordrecht (1996) 73–109. | MR | Zbl

G. Lebeau and E. Zuazua, Decay rates for the three-dimensional linear system of thermoelasticity. Arch. Ration Mech. Anal. 148 (1999) 179–231. | DOI | MR | Zbl

J.-L. Lions, Contrôlabilité exacte et stabilisation de systèmes distributés. Vol. 1 of Collection RMA. Masson, Paris (1988). | MR | Zbl

K. Liu, Locally distributed control and damping for the conservative systems. SIAM J. Control Optim. 35 (1997) 1574–1590. | DOI | MR | Zbl

W.-J. Liu and E. Zuazua, Decay rates for dissipative wave equations. Ricerche di Matematica 48 (1999) 61–75. | MR | Zbl

P. Martinez, A new method to obtain decay rate estimates for dissipative systems with localized damping. Rev. Mat. Complut. 12 (1999) 251–283. | DOI | MR | Zbl

P. Martinez, A new method to obtain decay rate estimates for dissipative systems. ESAIM: COCV 4 (1999) 419–444. | Numdam | MR | Zbl

P. Martinez and J. Vancostenoble, Exponential stability for the wave equation with weak nonmonotone damping. Portugal. Math. 57 (2000) 285–310. | MR | Zbl

M. Nakao, Decay of solutions of the wave equation with a local nonlinear dissipation. Math. Ann. 305 (1996) 403–417. | DOI | MR | Zbl

A. Soufyane, Stabilisation de la poutre de Timoshenko. C. R. Acad. Sci. Paris Sér I Math. 328 (1999) 731–734. | DOI | MR | Zbl

J. Vancostenoble, Optimalité d’estimation d’énergie pour une équation des ondes amortie. C. R. Acad. Sci. Paris série I 328 (1999) 777–782. | DOI | MR | Zbl

J. Vancostenoble and P. Martinez, Optimality of energy estimates for the wave equation with nonlinear boundary velocity feedbacks. SIAM J. Control Optim. 39 (2000) 776–797. | DOI | MR | Zbl

E. Zuazua, Uniform stabilization of the wave equation by nonlinear feedbacks. SIAM J. Control Optim. 28 (1989) 265–268. | MR | Zbl

E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping. Commun. Part. Differ. Equ. 15 (1990) 205–235. | DOI | MR | Zbl

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