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March  2016, 6(1): 1-25. doi: 10.3934/mcrf.2016.6.1

Local feedback stabilisation to a non-stationary solution for a damped non-linear wave equation

Kaïs Ammari 1, , Thomas Duyckaerts 2, and  Armen Shirikyan 3,

1. 

Département de Mathématiques, Faculté des Sciences de Monastir , Université de Monastir, 5019 Monastir, Tunisia
2. 

LAGA (UMR 7539), Institut Galilée, Université Paris 13, 99, avenue Jean-Baptiste Clément, 93430 Villetaneuse
3. 

Département de Mathématiques, Université de Cergy-Pontoise, UMR CNRS 8088, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise, France

Received  January 2015 Revised  October 2015 Published  January 2016

We study a damped semi-linear wave equation in a bounded domain of $\mathbb{R}^3$ with smooth boundary. It is proved that any $H^2$-smooth solution can be stabilised locally by a finite-dimensional feedback control supported by a given open subset satisfying a geometric condition. The proof is based on an investigation of the linearised equation, for which we construct a stabilising control satisfying the required properties. We next prove that the same control stabilises locally the non-linear problem.
Keywords: truncated observability inequality., distributed control, feedback stabilisation, Non-linear wave equation.
Mathematics Subject Classification: Primary: 35L71, 93B52; Secondary: 93B0.
Citation: Kaïs Ammari, Thomas Duyckaerts, Armen Shirikyan. Local feedback stabilisation to a non-stationary solution for a damped non-linear wave equation. Mathematical Control & Related Fields, 2016, 6 (1) : 1-25. doi: 10.3934/mcrf.2016.6.1
References:
[1]

C. Bardos, G. Lebeau and J. Rauch, Un exemple d'utilisation des notions de propagation pour le contrôle et la stabilisation de problèmes hyperboliques,, Rend. Sem. Mat. Univ. Politec. Torino (1988), (1988), 11.   Google Scholar

[2]

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.  doi: 10.1137/0330055.  Google Scholar

[3]

V. Barbu, S. S. Rodrigues and A. Shirikyan, Internal exponential stabilization to a non-stationary solution for 3D Navier-Stokes equations,, SIAM J. Control Optim., 49 (2011), 1454.  doi: 10.1137/100785739.  Google Scholar

[4]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, North-Holland Publishing, (1992).   Google Scholar

[5]

W. C. Chewning, Controllability of the nonlinear wave equation in several space variables,, SIAM J. Control Optim., 14 (1976), 19.  doi: 10.1137/0314002.  Google Scholar

[6]

J.-M. Coron and E. Trélat, Global steady-state stabilization and controllability of 1D semilinear wave equations,, Commun. Contemp. Math., 8 (2006), 535.  doi: 10.1142/S0219199706002209.  Google Scholar

[7]

B. Dehman, P. Gérard and G. Lebeau, Stabilization and control for the nonlinear Schrödinger equation on a compact surface,, Math. Z., 254 (2006), 729.  doi: 10.1007/s00209-006-0005-3.  Google Scholar

[8]

B. Dehman and G. Lebeau, Analysis of the HUM control operator and exact controllability for semilinear waves in uniform time,, SIAM J. Control Optim., 48 (2009), 521.  doi: 10.1137/070712067.  Google Scholar

[9]

B. Dehman, G. Lebeau and E. Zuazua, Stabilization and control for the subcritical semilinear wave equation,, Ann. Sci. École Norm. Sup. (4), 36 (2003), 525.  doi: 10.1016/S0012-9593(03)00021-1.  Google Scholar

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T. Duyckaerts, X. Zhang and E. Zuazua, On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 1.  doi: 10.1016/j.anihpc.2006.07.005.  Google Scholar

[11]

H. O. Fattorini, Local controllability of a nonlinear wave equation,, Math. Systems Theory, 9 (1975), 30.  doi: 10.1007/BF01698123.  Google Scholar

[12]

D. Fujiwara, Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order,, Proc. Japan Acad., 43 (1967), 82.  doi: 10.3792/pja/1195521686.  Google Scholar

[13]

X. Fu, J. Yong and X. Zhang, Exact controllability for multidimensional semilinear hyperbolic equations,, SIAM J. Control Optim., 46 (2007), 1578.  doi: 10.1137/040610222.  Google Scholar

[14]

A. Haraux, Two remarks on hyperbolic dissipative problems,, Nonlinear partial differential equations and their applications. Collège de France seminar, 122 (1985), 1983.   Google Scholar

[15]

L. Hörmander, The Analysis of Linear Partial Differential Operators. III,, Springer-Verlag, (1994).   Google Scholar

[16]

R. Joly and C. Laurent, A note on the semiglobal controllability of the semilinear wave equation,, SIAM J. Control Optim., 52 (2014), 439.  doi: 10.1137/120891174.  Google Scholar

[17]

C. Laurent, Global controllability and stabilization for the nonlinear Schrödinger equation on some compact manifolds of dimension 3,, SIAM J. Math. Anal., 42 (2010), 785.  doi: 10.1137/090749086.  Google Scholar

[18]

C. Laurent, On stabilization and control for the critical Klein-Gordon equation on a 3-D compact manifold,, J. Funct. Anal., 260 (2011), 1304.  doi: 10.1016/j.jfa.2010.10.019.  Google Scholar

[19]

J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,, Dunod, (1969).   Google Scholar

[20]

J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 1,, Masson, (1988).   Google Scholar

[21]

G. Lebeau and L. Robbiano, Stabilisation de l'équation des ondes par le bord,, Duke Math. J., 86 (1997), 465.  doi: 10.1215/S0012-7094-97-08614-2.  Google Scholar

[22]

L. Li and X. Zhang, Exact controllability for semilinear wave equations,, J. Math. Anal. Appl., 250 (2000), 589.  doi: 10.1006/jmaa.2000.6998.  Google Scholar

[23]

S. Zelik, Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent,, Commun. Pure Appl. Anal., 3 (2004), 921.  doi: 10.3934/cpaa.2004.3.921.  Google Scholar

[24]

X. Zhang, Exact controllability of semilinear evolution systems and its application,, J. Optim. Theory Appl., 107 (2000), 415.  doi: 10.1023/A:1026460831701.  Google Scholar

[25]

E. Zuazua, Exact controllability for the semilinear wave equation,, J. Math. Pures Appl. (9), 69 (1990), 1.   Google Scholar

[26]

E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping,, Comm. Partial Differential Equations, 15 (1990), 205.  doi: 10.1080/03605309908820684.  Google Scholar

[27]

E. Zuazua, Exact controllability for semilinear wave equations in one space dimension,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 109.   Google Scholar

show all references

References:
[1]

C. Bardos, G. Lebeau and J. Rauch, Un exemple d'utilisation des notions de propagation pour le contrôle et la stabilisation de problèmes hyperboliques,, Rend. Sem. Mat. Univ. Politec. Torino (1988), (1988), 11.   Google Scholar

[2]

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.  doi: 10.1137/0330055.  Google Scholar

[3]

V. Barbu, S. S. Rodrigues and A. Shirikyan, Internal exponential stabilization to a non-stationary solution for 3D Navier-Stokes equations,, SIAM J. Control Optim., 49 (2011), 1454.  doi: 10.1137/100785739.  Google Scholar

[4]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, North-Holland Publishing, (1992).   Google Scholar

[5]

W. C. Chewning, Controllability of the nonlinear wave equation in several space variables,, SIAM J. Control Optim., 14 (1976), 19.  doi: 10.1137/0314002.  Google Scholar

[6]

J.-M. Coron and E. Trélat, Global steady-state stabilization and controllability of 1D semilinear wave equations,, Commun. Contemp. Math., 8 (2006), 535.  doi: 10.1142/S0219199706002209.  Google Scholar

[7]

B. Dehman, P. Gérard and G. Lebeau, Stabilization and control for the nonlinear Schrödinger equation on a compact surface,, Math. Z., 254 (2006), 729.  doi: 10.1007/s00209-006-0005-3.  Google Scholar

[8]

B. Dehman and G. Lebeau, Analysis of the HUM control operator and exact controllability for semilinear waves in uniform time,, SIAM J. Control Optim., 48 (2009), 521.  doi: 10.1137/070712067.  Google Scholar

[9]

B. Dehman, G. Lebeau and E. Zuazua, Stabilization and control for the subcritical semilinear wave equation,, Ann. Sci. École Norm. Sup. (4), 36 (2003), 525.  doi: 10.1016/S0012-9593(03)00021-1.  Google Scholar

[10]

T. Duyckaerts, X. Zhang and E. Zuazua, On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 1.  doi: 10.1016/j.anihpc.2006.07.005.  Google Scholar

[11]

H. O. Fattorini, Local controllability of a nonlinear wave equation,, Math. Systems Theory, 9 (1975), 30.  doi: 10.1007/BF01698123.  Google Scholar

[12]

D. Fujiwara, Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order,, Proc. Japan Acad., 43 (1967), 82.  doi: 10.3792/pja/1195521686.  Google Scholar

[13]

X. Fu, J. Yong and X. Zhang, Exact controllability for multidimensional semilinear hyperbolic equations,, SIAM J. Control Optim., 46 (2007), 1578.  doi: 10.1137/040610222.  Google Scholar

[14]

A. Haraux, Two remarks on hyperbolic dissipative problems,, Nonlinear partial differential equations and their applications. Collège de France seminar, 122 (1985), 1983.   Google Scholar

[15]

L. Hörmander, The Analysis of Linear Partial Differential Operators. III,, Springer-Verlag, (1994).   Google Scholar

[16]

R. Joly and C. Laurent, A note on the semiglobal controllability of the semilinear wave equation,, SIAM J. Control Optim., 52 (2014), 439.  doi: 10.1137/120891174.  Google Scholar

[17]

C. Laurent, Global controllability and stabilization for the nonlinear Schrödinger equation on some compact manifolds of dimension 3,, SIAM J. Math. Anal., 42 (2010), 785.  doi: 10.1137/090749086.  Google Scholar

[18]

C. Laurent, On stabilization and control for the critical Klein-Gordon equation on a 3-D compact manifold,, J. Funct. Anal., 260 (2011), 1304.  doi: 10.1016/j.jfa.2010.10.019.  Google Scholar

[19]

J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,, Dunod, (1969).   Google Scholar

[20]

J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 1,, Masson, (1988).   Google Scholar

[21]

G. Lebeau and L. Robbiano, Stabilisation de l'équation des ondes par le bord,, Duke Math. J., 86 (1997), 465.  doi: 10.1215/S0012-7094-97-08614-2.  Google Scholar

[22]

L. Li and X. Zhang, Exact controllability for semilinear wave equations,, J. Math. Anal. Appl., 250 (2000), 589.  doi: 10.1006/jmaa.2000.6998.  Google Scholar

[23]

S. Zelik, Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent,, Commun. Pure Appl. Anal., 3 (2004), 921.  doi: 10.3934/cpaa.2004.3.921.  Google Scholar

[24]

X. Zhang, Exact controllability of semilinear evolution systems and its application,, J. Optim. Theory Appl., 107 (2000), 415.  doi: 10.1023/A:1026460831701.  Google Scholar

[25]

E. Zuazua, Exact controllability for the semilinear wave equation,, J. Math. Pures Appl. (9), 69 (1990), 1.   Google Scholar

[26]

E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping,, Comm. Partial Differential Equations, 15 (1990), 205.  doi: 10.1080/03605309908820684.  Google Scholar

[27]

E. Zuazua, Exact controllability for semilinear wave equations in one space dimension,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 109.   Google Scholar

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