Open Access
Issue
ESAIM: COCV
Volume 30, 2024
Article Number 38
Number of page(s) 26
DOI https://doi.org/10.1051/cocv/2024027
Published online 03 May 2024
  1. V. Komornik, Exact controllability and stabilization. RAM: Research in Applied Mathematics. Masson, Paris; John Wiley & Sons, Ltd., Chichester (1994). [Google Scholar]
  2. J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], Vol. 8. Masson, Paris (1988), Contrôlabilité exacte [Exact controllability], With appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch. [Google Scholar]
  3. 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. [Google Scholar]
  4. G. Lebeau, Damped wave equation. Algebraic and geometric methods in mathematical physics. Proceedings of the 1st Ukrainian–French–Romanian Summer School, Kaciveli, Ukraine, September 1–14, 1993. Kluwer Academic Publishers, Dordrecht (1996) 73–109 (in French). [Google Scholar]
  5. J. Rauch and M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domains. Indiana Univ. Math. J. 24 (1974) 79–86. [CrossRef] [MathSciNet] [Google Scholar]
  6. J. Le Rousseau, G. Lebeau, P. Terpolilli, and E. Trélat, Geometric control condition for the wave equation with a time-dependent observation domain. Anal. PDE 10 (2017) 983–1015. [CrossRef] [MathSciNet] [Google Scholar]
  7. C.M. Dafermos, Asymptotic behavior of solutions of evolution equations, Nonlinear evolution equations (Proc. Sympos., Univ. Wisconsin, Madison, Wis.), Publ. Math. Res. Center Univ. Wisconsin, Vol. 40. Academic Press, New York-London (1978) 103–123. [Google Scholar]
  8. A. Haraux, Comportement à l’infini pour une équation d’ondes non linéaire dissipative. C. R. Acad. Sci. Paris Sér. A-B 287 (1978) A507–A509. [Google Scholar]
  9. F. Alabau-Boussouira, Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems. Appl. Math. Optim. 51 (2005) 61–105. [Google Scholar]
  10. F. Alabau-Boussouira, New trends towards lower energy estimates and optimality for nonlinearly damped vibrating systems. J. Differ. Equ. 249 (2010) 1145–1178. [Google Scholar]
  11. Y. Chitour, S. Marx and C. Prieur, Lp-asymptotic stability analysis of a 1D wave equation with a nonlinear damping. J. Differ. Equ. 269 (2020) 8107–8131. [CrossRef] [Google Scholar]
  12. A. Haraux and E. Zuazua, Decay estimates for some semilinear damped hyperbolic problems. Arch. Ration. Mech. Anal. 100 (1988) 191–206 (in English). [CrossRef] [Google Scholar]
  13. P. Martinez, Stabilization for the wave equation with Neumann boundary condition by a locally distributed damping, Contrôle des systèmes gouvernés par des équations aux dérivées partielles (Nancy, 1999). ESAIM Proc., Vol. 8. Soc. Math. Appl. Indust., Paris (2000) 119–136. [CrossRef] [EDP Sciences] [Google Scholar]
  14. P. Martinez and J. Vancostenoble, Exponential stability for the wave equation with weak nonmonotone damping. Portugal. Math. 57 (2000) 285–310. [MathSciNet] [Google Scholar]
  15. E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping. Commun. Partial Differ. Equ. 15 (1990) 205–235. [CrossRef] [Google Scholar]
  16. J.P. Quinn and D.L. Russell, Asymptotic stability and energy decay rates for solutions of hyperbolic equations with boundary damping. Proc. Roy. Soc. Edinburgh Sect. A 77 (1977) 97–127. [CrossRef] [MathSciNet] [Google Scholar]
  17. J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation. J. Differ. Equ. 50 (1983) 163–182. [Google Scholar]
  18. K. Liu, Locally distributed control and damping for the conservative systems. SIAM J. Control Optim. 35 (1997) 09. [Google Scholar]
  19. F. Alabau-Boussouira, On some recent advances on stabilization for hyperbolic equations, in Control of Partial Differential Equations. Vol. 2048 of Lecture Notes in Mathematics, edited by Cannarsa, Piermarco, Coron and Jean-Michel. Springer (2012) 1–100. [Google Scholar]
  20. J.C. Peral, Lp estimates for the wave equation. J. Funct. Anal. 36 (1980) 114–145. [CrossRef] [Google Scholar]
  21. M. Kafnemer, B. Mebkhout, F. Jean and Y. Chitour, Lp-asymptotic stability of 1D damped wave equations with localized and linear damping. ESAIM Control Optim. Calc. Var. 28 (2022) 32. [CrossRef] [EDP Sciences] [Google Scholar]
  22. Y. Chitour, S. Marx and G. Mazanti, One-dimensional wave equation with set-valued boundary damping: well-posedness, asymptotic stability, and decay rates. ESAIM Control Optim. Calc. Var. 27 (2021) 62. [CrossRef] [EDP Sciences] [Google Scholar]
  23. J.-M. Coron and H.-M. Nguyen, Dissipative boundary conditions for nonlinear 1-D hyperbolic systems: sharp conditions through an approach via time-delay systems. SIAM J. Math. Anal. 47 (2015) 2220–2240. [CrossRef] [MathSciNet] [Google Scholar]
  24. J.-M. Coron and H.-M. Nguyen, Optimal time for the controllability of linear hyperbolic systems in one-dimensional space. SIAM J. Control Optim. 57 (2019) 1127–1156. [CrossRef] [MathSciNet] [Google Scholar]
  25. J.-M. Coron and H.-M. Nguyen, Finite-time stabilization in optimal time of homogeneous quasilinear hyperbolic systems in one dimensional space. ESAIM Control Optim. Calc. Var. 26 (2020) 24 [CrossRef] [EDP Sciences] [Google Scholar]
  26. J.-M. Coron and H.-M. Nguyen, Null-controllability of linear hyperbolic systems in one dimensional space. Syst. Control Lett. 148 (2021) 104851. [CrossRef] [Google Scholar]
  27. A. Haraux, lp estimates of solutions to some non-linear wave equations in one space dimension. Int. J. Math. Model. Numer. Optim. 1 (2009) 148–152. [Google Scholar]
  28. J.-M. Coron and H.-M. Nguyen, On the optimal controllability time for linear hyperbolic systems with time-dependent coefficients, https://arxiv.org/abs/2103.02653 (2021). [Google Scholar]
  29. D.L. Russell, A unified boundary controllability theory for hyperbolic and parabolic partial differential equations (1973). [Google Scholar]
  30. D.L. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions. SIAM Rev. 20 (1978) 639–739. [Google Scholar]
  31. G. Bastin and J.-M. Coron, Stability and boundary stabilization of 1-D hyperbolic systems, in Progress in Nonlinear Differential Equations and their Applications, Vol. 88. Birkhäuser/Springer, Cham (2016). [CrossRef] [Google Scholar]
  32. F. John and L. Nirenberg, On functions of bounded mean oscillation. Commun. Pure Appl. Math. 14 (1961) 415–426. [Google Scholar]
  33. H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext, Springer, New York (2011). [Google Scholar]
  34. G. Allaire, Analyse Numérique et Optimisation, 2nd edn. Édition de l’École Polytechnique (2005). [Google Scholar]
  35. J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, Vol. 136. American Mathematical Society, Providence, R, (2007). [Google Scholar]
  36. H.-M. Nguyen and M. Squassina, Fractional Caffarelli–Kohn–Nirenberg inequalities. J. Funct. Anal. 274 (2018) 2661–2672. [CrossRef] [MathSciNet] [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.