Buttazzo, G. ; Casas, E. ; de Teresa, L. ; Glowinski, R. ; Leugering, G. ; Trélat, E. ; Zhang, X.(éd.)
In this work we study the semi-discrete linearized Benjamin-Bona-Mahony equation (BBM) which is a model for propagation of one-dimensional, unidirectional, small amplitude long waves in non-linear dispersive media. In particular, we derive a stability estimate which yields a unique continuation property. The proof is based on a Carleman estimate for a finite difference approximation of Laplace operator with boundary observation in which the large parameter is connected to the mesh size.
Keywords: Benjamin-Bona-Mahony equation, unique continuation property, Carleman estimate discrete Carleman inequalities, dispersive equations, water wave equation, finite difference method, semi-discrete equations
@article{COCV_2021__27_1_A95_0,
author = {Lecaros, Rodrigo and Ortega, Jaime H. and P\'erez, Ariel},
editor = {Buttazzo, G. and Casas, E. and de Teresa, L. and Glowinski, R. and Leugering, G. and Tr\'elat, E. and Zhang, X.},
title = {Stability estimate for the semi-discrete linearized {Benjamin-Bona-Mahony} equation},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021087},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021087/}
}
TY - JOUR AU - Lecaros, Rodrigo AU - Ortega, Jaime H. AU - Pérez, Ariel ED - Buttazzo, G. ED - Casas, E. ED - de Teresa, L. ED - Glowinski, R. ED - Leugering, G. ED - Trélat, E. ED - Zhang, X. TI - Stability estimate for the semi-discrete linearized Benjamin-Bona-Mahony 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/2021087/ DO - 10.1051/cocv/2021087 LA - en ID - COCV_2021__27_1_A95_0 ER -
%0 Journal Article %A Lecaros, Rodrigo %A Ortega, Jaime H. %A Pérez, Ariel %E Buttazzo, G. %E Casas, E. %E de Teresa, L. %E Glowinski, R. %E Leugering, G. %E Trélat, E. %E Zhang, X. %T Stability estimate for the semi-discrete linearized Benjamin-Bona-Mahony 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/2021087/ %R 10.1051/cocv/2021087 %G en %F COCV_2021__27_1_A95_0
Lecaros, Rodrigo; Ortega, Jaime H.; Pérez, Ariel. Stability estimate for the semi-discrete linearized Benjamin-Bona-Mahony equation. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 93. doi: 10.1051/cocv/2021087
[1] and , Convergence of an inverse problem for a 1-D discrete wave equation. SIAM J. Control Optim. 51 (2013) 556–598.
[2] , and , Model equations for long waves in nonlinear dispersive systems. Philos. Trans. Roy. Soc. London Ser. A 272 (1972) 47–78.
[3] , and Discrete Carleman estimates for elliptic operators and uniform controllability of semi-discretized parabolic equations. J. Math. Pures Appl. 93 (2010) 240–276.
[4] and Carleman estimates for semi-discrete parabolic operators and application to the controllability of semi-linear semi-discrete parabolic equations. Ann. Inst. Henri Poincaré Anal. Non Linéaire 31 (2014) 1035–1078.
[5] , Sur un problème d’unicité pur les systèmes d’équations aux dérivées partielles à deux variables indépendantes. Ark. Mat., Astr. Fys. 26 (1939) 9.
[6] and , A geometrical demonstration for continuation of solutions of the generalised BBM equation. Monatshefte für Mathematik 194 (2021) 495–502.
[7] and Uniform stability estimates for the discrete Calderón problems. Inverse Probl. 27 (2011) 125012.
[8] , and , Carleman estimates for second order partial differential operators and applications. SpringerBriefs in Mathematics. Springer, Cham (2019). A unified approach, BCAM SpringerBriefs.
[9] and , Controllability of evolution equations. Vol. 34 of Lecture Notes Series. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul (1996).
[10] and Carleman estimates and controllability results for fully-discrete approximations of 1-d parabolic equations. Preprint (2020). | arXiv
[11] , Inverse source problems. Vol. 34 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (1990).
[12] , On the controllability of the linearized Benjamin-Bona-Mahony equation. SIAM J. Control Optim. 39 (2001) 1677–1696.
[13] , Carleman estimates for semi-discrete parabolic operators with a discontinuous diffusion coefficient and applications to controllability. Math. Control Relat. Fields 4 (2014) 203–259.
[14] and , Unique continuation property and control for the Benjamin-Bona-Mahony equation on a periodic domain. J. Differ. Equ. 254 (2013) 141–178.
[15] , One unique continuation for a linearized Benjamin-Bona-Mahony equation. J. Inverse Ill-Posed Probl. 11 (2003) 537–543.
[16] and , Unique continuation for the linearized Benjamin-Bona-Mahony equation with space-dependent potential. Math. Ann. 325 (2003) 543–582.
[17] , Inverse problems for the fourth order Schrödinger equation on a finite domain. Math. Control Relat. Fields 5 (2015) 177–189.
[18] , Propagation, observation, and control of waves approximated by finite difference methods. SIAM Rev. 47 (2005) 197–243.
Cité par Sources :
This work is dedicated to Prof. Enrique Zuazua. Dear Enrique, thanks for these years of friendship and for your valuable contributions to the study of control and partial differential equations, and also, for the formation of several generations of researchers in Latin America.
A. Pérez was founded by the National Agency for Research and Development (ANID)/Scholarship Program/ Doctorado Nacional Chile/2017 – 21170495. R. Lecaros was partially supported by FONDECYT(Chile) Grant 11180874. J.H. Ortega was partially supported by Centro de Modelamiento Matemático (AFB170001) and FONDECYT(Chile) Grant 1201125.
