American Institute of Mathematical Sciences

Advanced
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

[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

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

[1]

Xu Zhang, Chuang Zheng, Enrique Zuazua. Time discrete wave equations: Boundary observability and control. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 571-604. doi: 10.3934/dcds.2009.23.571
[2]

Denis Serre. Non-linear electromagnetism and special relativity. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 435-454. doi: 10.3934/dcds.2009.23.435
[3]

Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302
[4]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444
[5]

Mokhtari Yacine. Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021004
[6]

Ahmad El Hajj, Hassan Ibrahim, Vivian Rizik. $ BV $ solution for a non-linear Hamilton-Jacobi system. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020405
[7]

Ludovick Gagnon, José M. Urquiza. Uniform boundary observability with Legendre-Galerkin formulations of the 1-D wave equation. Evolution Equations & Control Theory, 2021, 10 (1) : 129-153. doi: 10.3934/eect.2020054
[8]

Christian Clason, Vu Huu Nhu, Arnd Rösch. Optimal control of a non-smooth quasilinear elliptic equation. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020052
[9]

Hyung-Chun Lee. Efficient computations for linear feedback control problems for target velocity matching of Navier-Stokes flows via POD and LSTM-ROM. Electronic Research Archive, , () : -. doi: 10.3934/era.2020128
[10]

Touria Karite, Ali Boutoulout. Global and regional constrained controllability for distributed parabolic linear systems: RHUM approach. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020055
[11]

Feifei Cheng, Ji Li. Geometric singular perturbation analysis of Degasperis-Procesi equation with distributed delay. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 967-985. doi: 10.3934/dcds.2020305
[12]

Yuan Tan, Qingyuan Cao, Lan Li, Tianshi Hu, Min Su. A chance-constrained stochastic model predictive control problem with disturbance feedback. Journal of Industrial & Management Optimization, 2021, 17 (1) : 67-79. doi: 10.3934/jimo.2019099
[13]

Yanan Li, Zhijian Yang, Na Feng. Uniform attractors and their continuity for the non-autonomous Kirchhoff wave models. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021018
[14]

Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021015
[15]

Bopeng Rao, Zhuangyi Liu. A spectral approach to the indirect boundary control of a system of weakly coupled wave equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 399-414. doi: 10.3934/dcds.2009.23.399
[16]

Guangjun Shen, Xueying Wu, Xiuwei Yin. Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 755-774. doi: 10.3934/dcdsb.2020133
[17]

Biao Zeng. Existence results for fractional impulsive delay feedback control systems with Caputo fractional derivatives. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021001
[18]

Xinyu Mei, Yangmin Xiong, Chunyou Sun. Pullback attractor for a weakly damped wave equation with sup-cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 569-600. doi: 10.3934/dcds.2020270
[19]

Takiko Sasaki. Convergence of a blow-up curve for a semilinear wave equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1133-1143. doi: 10.3934/dcdss.2020388
[20]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296

2019 Impact Factor: 0.857

Tools

Metrics

  • PDF downloads (49)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences