# American Institute of Mathematical Sciences

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

 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.
Citation: Kaïs Ammari, Thomas Duyckaerts, Armen Shirikyan. Local feedback stabilisation to a non-stationary solution for a damped non-linear wave equation. Mathematical Control and 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), Special Issue, 11-31 (1989). [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), 1024-1065. doi: 10.1137/0330055. [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), 1454-1478. doi: 10.1137/100785739. [4] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland Publishing, Amsterdam, 1992. [5] W. C. Chewning, Controllability of the nonlinear wave equation in several space variables, SIAM J. Control Optim., 14 (1976), 19-25. doi: 10.1137/0314002. [6] J.-M. Coron and E. Trélat, Global steady-state stabilization and controllability of 1D semilinear wave equations, Commun. Contemp. Math., 8 (2006), 535-567. doi: 10.1142/S0219199706002209. [7] B. Dehman, P. Gérard and G. Lebeau, Stabilization and control for the nonlinear Schrödinger equation on a compact surface, Math. Z., 254 (2006), 729-749. doi: 10.1007/s00209-006-0005-3. [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), 521-550. doi: 10.1137/070712067. [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), 525-551. doi: 10.1016/S0012-9593(03)00021-1. [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), 1-41. doi: 10.1016/j.anihpc.2006.07.005. [11] H. O. Fattorini, Local controllability of a nonlinear wave equation, Math. Systems Theory, 9 (1975), 30-45. doi: 10.1007/BF01698123. [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), 82-86. doi: 10.3792/pja/1195521686. [13] X. Fu, J. Yong and X. Zhang, Exact controllability for multidimensional semilinear hyperbolic equations, SIAM J. Control Optim., 46 (2007), 1578-1614 (electronic). doi: 10.1137/040610222. [14] A. Haraux, Two remarks on hyperbolic dissipative problems, Nonlinear partial differential equations and their applications. Collège de France seminar, Vol. VII (Paris, 1983-1984), Res. Notes in Math., Pitman, Boston, MA, 122 (1985), 161-179. [15] L. Hörmander, The Analysis of Linear Partial Differential Operators. III, Springer-Verlag, Berlin, 1994. [16] R. Joly and C. Laurent, A note on the semiglobal controllability of the semilinear wave equation, SIAM J. Control Optim., 52 (2014), 439-450. doi: 10.1137/120891174. [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), 785-832. doi: 10.1137/090749086. [18] C. Laurent, On stabilization and control for the critical Klein-Gordon equation on a 3-D compact manifold, J. Funct. Anal., 260 (2011), 1304-1368. doi: 10.1016/j.jfa.2010.10.019. [19] J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, 1969. [20] J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 1, Masson, Paris, 1988. [21] G. Lebeau and L. Robbiano, Stabilisation de l'équation des ondes par le bord, Duke Math. J., 86 (1997), 465-491. doi: 10.1215/S0012-7094-97-08614-2. [22] L. Li and X. Zhang, Exact controllability for semilinear wave equations, J. Math. Anal. Appl., 250 (2000), 589-597. doi: 10.1006/jmaa.2000.6998. [23] S. Zelik, Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent, Commun. Pure Appl. Anal., 3 (2004), 921-934. doi: 10.3934/cpaa.2004.3.921. [24] X. Zhang, Exact controllability of semilinear evolution systems and its application, J. Optim. Theory Appl., 107 (2000), 415-432. doi: 10.1023/A:1026460831701. [25] E. Zuazua, Exact controllability for the semilinear wave equation, J. Math. Pures Appl. (9), 69 (1990), 1-31. [26] E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping, Comm. Partial Differential Equations, 15 (1990), 205-235. doi: 10.1080/03605309908820684. [27] E. Zuazua, Exact controllability for semilinear wave equations in one space dimension, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 109-129.

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##### 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), Special Issue, 11-31 (1989). [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), 1024-1065. doi: 10.1137/0330055. [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), 1454-1478. doi: 10.1137/100785739. [4] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland Publishing, Amsterdam, 1992. [5] W. C. Chewning, Controllability of the nonlinear wave equation in several space variables, SIAM J. Control Optim., 14 (1976), 19-25. doi: 10.1137/0314002. [6] J.-M. Coron and E. Trélat, Global steady-state stabilization and controllability of 1D semilinear wave equations, Commun. Contemp. Math., 8 (2006), 535-567. doi: 10.1142/S0219199706002209. [7] B. Dehman, P. Gérard and G. Lebeau, Stabilization and control for the nonlinear Schrödinger equation on a compact surface, Math. Z., 254 (2006), 729-749. doi: 10.1007/s00209-006-0005-3. [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), 521-550. doi: 10.1137/070712067. [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), 525-551. doi: 10.1016/S0012-9593(03)00021-1. [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), 1-41. doi: 10.1016/j.anihpc.2006.07.005. [11] H. O. Fattorini, Local controllability of a nonlinear wave equation, Math. Systems Theory, 9 (1975), 30-45. doi: 10.1007/BF01698123. [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), 82-86. doi: 10.3792/pja/1195521686. [13] X. Fu, J. Yong and X. Zhang, Exact controllability for multidimensional semilinear hyperbolic equations, SIAM J. Control Optim., 46 (2007), 1578-1614 (electronic). doi: 10.1137/040610222. [14] A. Haraux, Two remarks on hyperbolic dissipative problems, Nonlinear partial differential equations and their applications. Collège de France seminar, Vol. VII (Paris, 1983-1984), Res. Notes in Math., Pitman, Boston, MA, 122 (1985), 161-179. [15] L. Hörmander, The Analysis of Linear Partial Differential Operators. III, Springer-Verlag, Berlin, 1994. [16] R. Joly and C. Laurent, A note on the semiglobal controllability of the semilinear wave equation, SIAM J. Control Optim., 52 (2014), 439-450. doi: 10.1137/120891174. [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), 785-832. doi: 10.1137/090749086. [18] C. Laurent, On stabilization and control for the critical Klein-Gordon equation on a 3-D compact manifold, J. Funct. Anal., 260 (2011), 1304-1368. doi: 10.1016/j.jfa.2010.10.019. [19] J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, 1969. [20] J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 1, Masson, Paris, 1988. [21] G. Lebeau and L. Robbiano, Stabilisation de l'équation des ondes par le bord, Duke Math. J., 86 (1997), 465-491. doi: 10.1215/S0012-7094-97-08614-2. [22] L. Li and X. Zhang, Exact controllability for semilinear wave equations, J. Math. Anal. Appl., 250 (2000), 589-597. doi: 10.1006/jmaa.2000.6998. [23] S. Zelik, Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent, Commun. Pure Appl. Anal., 3 (2004), 921-934. doi: 10.3934/cpaa.2004.3.921. [24] X. Zhang, Exact controllability of semilinear evolution systems and its application, J. Optim. Theory Appl., 107 (2000), 415-432. doi: 10.1023/A:1026460831701. [25] E. Zuazua, Exact controllability for the semilinear wave equation, J. Math. Pures Appl. (9), 69 (1990), 1-31. [26] E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping, Comm. Partial Differential Equations, 15 (1990), 205-235. doi: 10.1080/03605309908820684. [27] E. Zuazua, Exact controllability for semilinear wave equations in one space dimension, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 109-129.
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