Open Access
Issue
ESAIM: COCV
Volume 27, 2021
Article Number 13
Number of page(s) 32
DOI https://doi.org/10.1051/cocv/2021010
Published online 19 March 2021
  1. H. Banks, W. Fang, R. Silcox and R. Smith, Approximation methods for control of structural acoustics models with piezoceramic actuators. J. Intell. Mater. Syst. Struct. 4 (1993) 98–116. [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) 1024–1065. [Google Scholar]
  3. A. Bottois, Pointwise moving control for the 1-D wave equation – Numerical approximation and optimization of the support. To appear in Radon Series on Computational and Applied Mathematics. De Gruyter, In press. https://hal.archives-ouvertes.fr/hal-02972968 (2021). [Google Scholar]
  4. A.E. Brouwer and W.H. Haemers, Spectra of graphs, Universitext. Springer, New York (2012). [CrossRef] [Google Scholar]
  5. C. Castro, Exact controllability of the 1-D wave equation from a moving interior point. ESAIM: COCV 19 (2013) 301–316. [CrossRef] [EDP Sciences] [Google Scholar]
  6. C. Castro, N. Cîndea and A. Münch, Controllability of the linear one-dimensional wave equation with inner moving forces. SIAM J. Control Optim. 52 (2014) 4027–4056. [Google Scholar]
  7. D. Chenais, On the existence of a solution in a domain identification problem. J. Math. Anal. Appl. 52 (1975) 189–219. [Google Scholar]
  8. F.R.K. Chung, Spectral graph theory. Vol. 92 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI (1997). [Google Scholar]
  9. N. Cîndea and A. Münch, A mixed formulation for the direct approximation of the control of minimal L2-norm for linear type wave equations. Calcolo 52 (2015) 245–288. [Google Scholar]
  10. J.-M. Coron, Control and nonlinearity. Vol. 136 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (2007). [Google Scholar]
  11. L. Cui, X. Liu and H. Gao, Exact controllability for a one-dimensional wave equation in non-cylindrical domains. J. Math. Anal. Appl. 402 (2013) 612–625. [Google Scholar]
  12. P. Destuynder, I. Legrain, L. Castel and N. Richard, Theoretical, numerical and experimental discussion on the use of piezoelectric devices for control–structure interaction. Eur. J. Mech. A. Solids 11 (1992) 181–213. [Google Scholar]
  13. B.H. Haak and D.-T. Hoang, Exact observability of a 1-dimensional wave equation on a noncylindrical domain. SIAM J. Control Optim. 57 (2019) 570–589. [Google Scholar]
  14. F. Hecht, New development in Freefem++. J. Numer. Math. 20 (2012) 251–265. [Google Scholar]
  15. A. Henrot and M. Pierre, Variation et optimisation de formes. Une analyse géométrique. [A geometric analysis]. Vol. 48 of Mathématiques & Applications (Berlin) [Mathematics & Applications]. Springer, Berlin (2005). [Google Scholar]
  16. A.Y. Khapalov, Controllability of the wave equation with moving point control. Appl. Math. Optim. 31 (1995) 155–175. [Google Scholar]
  17. J. Le Rousseau, G. Lebeau, P. Terpolilli and E. Trélat, Geometric control condition for the wave equation with a time-dependent observation domain. Anal. Partial Differ. Equ. 10 (2017) 983–1015. [Google Scholar]
  18. J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Contrôlabilité exacte. [Exact controllability], With appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch. Tome 1. Vol. 8 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics]. Masson, Paris (1988). [Google Scholar]
  19. K. Liu and J. Yong, Rapid exact controllability of the wave equation by controls distributed on a time-variant subdomain. Chin. Ann. Math. Ser. B 20 (1999) 65–76. A Chinese summary appears in Chin. Ann. Math. Ser. A 20 (1999) 142. [CrossRef] [Google Scholar]
  20. K.A. Lurie, An introduction to the mathematical theory of dynamic materials. Vol. 15 of Advances in Mechanics and Mathematics. Second edition, Springer, Cham (2017) MR2305885. [Google Scholar]
  21. P. Martin, L. Rosier and P. Rouchon, Null controllability of the structurally damped wave equation with moving control. SIAM J. Control Optim. 51 (2013) 660–684. [Google Scholar]
  22. B. Mohar, The Laplacian spectrum of graphs. Vol. 2 of Graph theory, combinatorics, and applications. (Kalamazoo, MI, 1988). Wiley-Intersci. Publ., Wiley, New York (1991) 871–898. [Google Scholar]
  23. A. Münch, Optimal design of the support of the control for the 2-D wave equation: a numerical method. Int. J. Numer. Anal. Model. 5 (2008) 331–351. [Google Scholar]
  24. A. Münch, Optimal location of the support of the control for the 1-D wave equation: numerical investigations. Comput. Optim. Appl. 42 (2009) 443–470. [Google Scholar]
  25. A. Münch, Numerical estimations of the cost of boundary controls for the equation yt εyxx + Myx = 0 with respect to ε, in Recent advances in PDEs: analysis, numerics and control. Vol. 17 of SEMA SIMAI Springer Ser. Springer, Cham (2018) 159–191. [CrossRef] [Google Scholar]
  26. A. Münch, P. Pedregal and F. Periago, Optimal design of the damping set for the stabilization of the wave equation. J. Differ. Equ. 231 (2006) 331–358. [Google Scholar]
  27. A.O. Özer, Potential formulation for charge or current-controlled piezoelectric smart composites and stabilization results: electrostatic versus quasi-static versus fully-dynamic approaches. IEEE Trans. Automat. Control 64 (2019) 989–1002. [Google Scholar]
  28. A.O. Özer and K.A. Morris, Modeling and stabilization of current-controlled piezo-electric beams with dynamic electromagnetic field. ESAIM:COCV 26 (2020) 24. [CrossRef] [EDP Sciences] [Google Scholar]
  29. F. Periago, Optimal shape and position of the support for the internal exact control of a string. Systems Control Lett. 58 (2009) 136–140. [CrossRef] [MathSciNet] [Google Scholar]
  30. Y. Privat, E. Trélat and E. Zuazua, Optimal location of controllers for the one-dimensional wave equation. Ann. Inst. Henri Poincaré Anal. Non Linéaire 30 (2013) 1097–1126. [CrossRef] [Google Scholar]
  31. Y. Privat, E. Trélat and E. Zuazua, Optimal observation of the one-dimensional wave equation. J. Fourier Anal. Appl. 19 (2013) 514–544. [Google Scholar]
  32. A. Shao, On Carleman and observability estimates for wave equations on time-dependent domains. Proc. Lond. Math. Soc. 119 (2019) 998–1064. [CrossRef] [Google Scholar]
  33. M. Tucsnak, Control of plate vibrations by means of piezoelectric actuators. Discrete Contin. Dynam. Syst. 2 (1996) 281–293. [CrossRef] [Google Scholar]

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