This work is concerned with the null controllability of the one-dimensional wave equation over non-cylindrical distributed domains. The controllability in that case has been obtained by Castro et al. [SIAM J. Control Optim. 52 (2014)] for domains satisfying the usual geometric optic condition. We analyze the problem of optimizing the non-cylindrical support q of the control of minimal L2(q)-norm. In this respect, we prove a uniform observability inequality for a class of domains q satisfying the geometric optic condition. The proof based on the d’Alembert formula relies on arguments from graph theory. Numerical experiments are discussed and highlight the influence of the initial condition on the optimal domains.
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Keywords: Wave equation, Uniform observability, Optimal shape
@article{COCV_2021__27_1_A15_0,
author = {Bottois, Arthur and C{\^\i}ndea, Nicolae and M\"unch, Arnaud},
title = {Optimization of non-cylindrical domains for the exact null controllability of the {1D} wave equation},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021010},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021010/}
}
TY - JOUR AU - Bottois, Arthur AU - Cîndea, Nicolae AU - Münch, Arnaud TI - Optimization of non-cylindrical domains for the exact null controllability of the 1D wave equation JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021010/ DO - 10.1051/cocv/2021010 LA - en ID - COCV_2021__27_1_A15_0 ER -
%0 Journal Article %A Bottois, Arthur %A Cîndea, Nicolae %A Münch, Arnaud %T Optimization of non-cylindrical domains for the exact null controllability of the 1D wave equation %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021010/ %R 10.1051/cocv/2021010 %G en %F COCV_2021__27_1_A15_0
Bottois, Arthur; Cîndea, Nicolae; Münch, Arnaud. Optimization of non-cylindrical domains for the exact null controllability of the 1D wave equation. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 13. doi: 10.1051/cocv/2021010
[1] , , and , Approximation methods for control of structural acoustics models with piezoceramic actuators. J. Intell. Mater. Syst. Struct. 4 (1993) 98–116.
[2] , and , Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J. Control Optim. 30 (1992) 1024–1065.
[3] , 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. (2021). | HAL
[4] and , Spectra of graphs, Universitext. Springer, New York (2012).
[5] , Exact controllability of the 1-D wave equation from a moving interior point. ESAIM: COCV 19 (2013) 301–316.
[6] , and , Controllability of the linear one-dimensional wave equation with inner moving forces. SIAM J. Control Optim. 52 (2014) 4027–4056.
[7] , On the existence of a solution in a domain identification problem. J. Math. Anal. Appl. 52 (1975) 189–219.
[8] , 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).
[9] and , A mixed formulation for the direct approximation of the control of minimal L2-norm for linear type wave equations. Calcolo 52 (2015) 245–288.
[10] , Control and nonlinearity. Vol. 136 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (2007).
[11] , and , Exact controllability for a one-dimensional wave equation in non-cylindrical domains. J. Math. Anal. Appl. 402 (2013) 612–625.
[12] , , and , Theoretical, numerical and experimental discussion on the use of piezoelectric devices for control–structure interaction. Eur. J. Mech. A. Solids 11 (1992) 181–213.
[13] and , Exact observability of a 1-dimensional wave equation on a noncylindrical domain. SIAM J. Control Optim. 57 (2019) 570–589.
[14] , New development in Freefem++. J. Numer. Math. 20 (2012) 251–265.
[15] and , Variation et optimisation de formes. Une analyse géométrique. [A geometric analysis]. Vol. 48 of Mathématiques & Applications (Berlin) [Mathematics & Applications]. Springer, Berlin (2005).
[16] , Controllability of the wave equation with moving point control. Appl. Math. Optim. 31 (1995) 155–175.
[17] , , and , Geometric control condition for the wave equation with a time-dependent observation domain. Anal. Partial Differ. Equ. 10 (2017) 983–1015.
[18] , 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).
[19] and , 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.
[20] , An introduction to the mathematical theory of dynamic materials. Vol. 15 of Advances in Mechanics and Mathematics. Second edition, Springer, Cham (2017) MR2305885.
[21] , and , Null controllability of the structurally damped wave equation with moving control. SIAM J. Control Optim. 51 (2013) 660–684.
[22] , 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.
[23] , 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.
[24] , Optimal location of the support of the control for the 1-D wave equation: numerical investigations. Comput. Optim. Appl. 42 (2009) 443–470.
[25] , 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.
[26] , and , Optimal design of the damping set for the stabilization of the wave equation. J. Differ. Equ. 231 (2006) 331–358.
[27] , 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.
[28] and , Modeling and stabilization of current-controlled piezo-electric beams with dynamic electromagnetic field. ESAIM:COCV 26 (2020) 24.
[29] , Optimal shape and position of the support for the internal exact control of a string. Systems Control Lett. 58 (2009) 136–140.
[30] , and , Optimal location of controllers for the one-dimensional wave equation. Ann. Inst. Henri Poincaré Anal. Non Linéaire 30 (2013) 1097–1126.
[31] , and , Optimal observation of the one-dimensional wave equation. J. Fourier Anal. Appl. 19 (2013) 514–544.
[32] , On Carleman and observability estimates for wave equations on time-dependent domains. Proc. Lond. Math. Soc. 119 (2019) 998–1064.
[33] , Control of plate vibrations by means of piezoelectric actuators. Discrete Contin. Dynam. Syst. 2 (1996) 281–293.
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