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## Exponential stabilization of solutions for the 1-d transmission wave equation with boundary feedback

#### Medjahed Djilali [1] , Ali Hakem [2]

##### 36 71

The purpose of this work is to study the exponential decay of the energy for
the one-dimensional transmission wave equation with a boundary velocity feedback.
Thanks to the perturbed energy method developed by some authors in several contexts, and
under certain conditions, we prove that the feedback controller exponentially stabilizes the
equilibrium to zero of the system below, i.e. the feedback leads to faster energy decay.
Boundary feedback, decay rate of energy, exponential stabilization, perturbed energy
• [1] K. Ammari, Derichlet boundary stabilization of the wave equation , Asymptot. Anal.30 (2002) 117-130.
• [2] G. Chen, Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain . J. Math. Pures Appl. 58, 249-273 (1979)
• [3] G. Chen, Control and stabilization for the wave equation in a bounded domain . SIAM J. Control Optim. 17, 66-81 (1979).
• [4] G. Chen, Control and stabilization for the wave equation, part III: Domain with moving boundary. SIAM J. Control Optim.19, 123-138 (1981).
• [5] C. Deng,Y. Liu, W. Jiang, F. Huang, Exponential decay rate for a wave equation with Dirichlet boundary control, Applied Mathematics letters, 20 (2007) 861-865.
• [6] L.C. Evans, Partial Differential Equations, Vol. 19, American Mathematical Society, 1997.
• [7] I. Lasiecka& R. Trigiani, Uniform exponential energy decay of the wave equation in a bounded region with feedback control in the Dirichlet boundary conditions, J. Differential Equations. 66 (1987) 340-390.
• [8] J.L. Lions, Controlabilite exacte perturbation et stabilisation de systemes distribues, Tome 1, Controlabilite exacte. Masson, Paris (1988).
• [9] J.L. Lions, Controlabilit´e exacte perturbation et stabilisation de systemes distribues, Tome 2, Perturbation. Masson, Paris (1988).
• [10] W. Liu, Stabilization and controllability for the transmission wave equation. IEEE Transcation on Automatic Control 46, 1900-1907 (2001).
• [11] W. Liu, E. Zuazua, Decay rates for dissipative wave equations. Ricerche di Matimatica. 48, 61-75 (1999).
• [12] W. Liu, E. Zuazua, Uniform stabilization of the higher dimensional system of thermoelastisity with boundary feedback. Quartyely Appl. Math. 59, 269-314 (2001).
• [13] M. Nakao, Energy decay for the wave equation with nonlinear weak dissipation. Differential Integral Equation, 8, 681-688 (1995).
• [14] J. Rauch& M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domains. India J. Math. 24, 79-83 (1974)
• [15] B. Straughan, The Energy Method, Stability, and Nonlinear Convection, Springer Science Business Media, New York, 2004.
• [16] E. Zuazua, Uniform stabilization of the wave equation by nonlinear boundary feedback. SIAM J. Control and optim. 28 (1990) 466-478.
• [17] E. Zuazua, Exponential decay for the semi-linear wave equation with locally distributed damping. Commun. in Partial Differential Equations 15, 205-235 (1990)
Primary Language en Mathematics December 2018 Articles Author: Medjahed Djilali (Primary Author)Institution: Department of Mathematics, Djillali Liabes University, 22000 SIDI-BEL-ABBES, Algeria.Country: Algeria Author: Ali HakemInstitution: Djillali Liabes University Publication Date: December 24, 2018
 Bibtex @research article { atnaa418379, journal = {Advances in the Theory of Nonlinear Analysis and its Application}, issn = {}, eissn = {2587-2648}, address = {Erdal KARAPINAR}, year = {2018}, volume = {2}, pages = {217 - 223}, doi = {10.31197/atnaa.418379}, title = {Exponential stabilization of solutions for the 1-d transmission wave equation with boundary feedback}, key = {cite}, author = {Djilali, Medjahed and Hakem, Ali} } APA Djilali, M , Hakem, A . (2018). Exponential stabilization of solutions for the 1-d transmission wave equation with boundary feedback. Advances in the Theory of Nonlinear Analysis and its Application, 2 (4), 217-223. DOI: 10.31197/atnaa.418379 MLA Djilali, M , Hakem, A . "Exponential stabilization of solutions for the 1-d transmission wave equation with boundary feedback". Advances in the Theory of Nonlinear Analysis and its Application 2 (2018): 217-223 Chicago Djilali, M , Hakem, A . "Exponential stabilization of solutions for the 1-d transmission wave equation with boundary feedback". Advances in the Theory of Nonlinear Analysis and its Application 2 (2018): 217-223 RIS TY - JOUR T1 - Exponential stabilization of solutions for the 1-d transmission wave equation with boundary feedback AU - Medjahed Djilali , Ali Hakem Y1 - 2018 PY - 2018 N1 - doi: 10.31197/atnaa.418379 DO - 10.31197/atnaa.418379 T2 - Advances in the Theory of Nonlinear Analysis and its Application JF - Journal JO - JOR SP - 217 EP - 223 VL - 2 IS - 4 SN - -2587-2648 M3 - doi: 10.31197/atnaa.418379 UR - https://doi.org/10.31197/atnaa.418379 Y2 - 2018 ER - EndNote %0 Advances in the Theory of Nonlinear Analysis and its Application Exponential stabilization of solutions for the 1-d transmission wave equation with boundary feedback %A Medjahed Djilali , Ali Hakem %T Exponential stabilization of solutions for the 1-d transmission wave equation with boundary feedback %D 2018 %J Advances in the Theory of Nonlinear Analysis and its Application %P -2587-2648 %V 2 %N 4 %R doi: 10.31197/atnaa.418379 %U 10.31197/atnaa.418379 ISNAD Djilali, Medjahed , Hakem, Ali . "Exponential stabilization of solutions for the 1-d transmission wave equation with boundary feedback". Advances in the Theory of Nonlinear Analysis and its Application 2 / 4 (December 2018): 217-223. https://doi.org/10.31197/atnaa.418379 AMA Djilali M , Hakem A . Exponential stabilization of solutions for the 1-d transmission wave equation with boundary feedback. ATNAA. 2018; 2(4): 217-223. Vancouver Djilali M , Hakem A . Exponential stabilization of solutions for the 1-d transmission wave equation with boundary feedback. Advances in the Theory of Nonlinear Analysis and its Application. 2018; 2(4): 223-217.