The dynamical Lame system : regularity of solutions, boundary controllability and boundary data continuation

M. I. Belishev; I. Lasiecka

ESAIM: Control, Optimisation and Calculus of Variations (2002)

  • Volume: 8, page 143-167
  • ISSN: 1292-8119

Abstract

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The boundary control problem for the dynamical Lame system (isotropic elasticity model) is considered. The continuity of the “input → state” map in L 2 -norms is established. A structure of the reachable sets for arbitrary T > 0 is studied. In general case, only the first component u ( · , T ) of the complete state { u ( · , T ) , u t ( · , T ) } may be controlled, an approximate controllability occurring in the subdomain filled with the shear (slow) waves. The controllability results are applied to the problem of the boundary data continuation. If T 0 exceeds the time needed for shear waves to fill the entire domain, then the response operator (“input → output” map) R 2 T 0 uniquely determines R T for any T > 0 . A procedure recovering R ∞ via R 2 T 0 is also described.

How to cite

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Belishev, M. I., and Lasiecka, I.. "The dynamical Lame system : regularity of solutions, boundary controllability and boundary data continuation." ESAIM: Control, Optimisation and Calculus of Variations 8 (2002): 143-167. <http://eudml.org/doc/245575>.

@article{Belishev2002,
abstract = {The boundary control problem for the dynamical Lame system (isotropic elasticity model) is considered. The continuity of the “input $\rightarrow $ state” map in $L_2$-norms is established. A structure of the reachable sets for arbitrary $T&gt;0$ is studied. In general case, only the first component $u(\cdot ,T)$ of the complete state $\lbrace u(\cdot ,T),u_t(\cdot ,T)\rbrace $ may be controlled, an approximate controllability occurring in the subdomain filled with the shear (slow) waves. The controllability results are applied to the problem of the boundary data continuation. If $T_0$ exceeds the time needed for shear waves to fill the entire domain, then the response operator (“input $\rightarrow $ output” map) $R^\{2T_0\}$ uniquely determines $R^T$ for any $T&gt;0$. A procedure recovering $R^\infty $ via $R^\{2T_0\}$ is also described.},
author = {Belishev, M. I., Lasiecka, I.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {isotropic elasticity; dynamical Lame system; regularity of solutions; structure of sets reachable from the boundary in a short time; boundary controllability},
language = {eng},
pages = {143-167},
publisher = {EDP-Sciences},
title = {The dynamical Lame system : regularity of solutions, boundary controllability and boundary data continuation},
url = {http://eudml.org/doc/245575},
volume = {8},
year = {2002},
}

TY - JOUR
AU - Belishev, M. I.
AU - Lasiecka, I.
TI - The dynamical Lame system : regularity of solutions, boundary controllability and boundary data continuation
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2002
PB - EDP-Sciences
VL - 8
SP - 143
EP - 167
AB - The boundary control problem for the dynamical Lame system (isotropic elasticity model) is considered. The continuity of the “input $\rightarrow $ state” map in $L_2$-norms is established. A structure of the reachable sets for arbitrary $T&gt;0$ is studied. In general case, only the first component $u(\cdot ,T)$ of the complete state $\lbrace u(\cdot ,T),u_t(\cdot ,T)\rbrace $ may be controlled, an approximate controllability occurring in the subdomain filled with the shear (slow) waves. The controllability results are applied to the problem of the boundary data continuation. If $T_0$ exceeds the time needed for shear waves to fill the entire domain, then the response operator (“input $\rightarrow $ output” map) $R^{2T_0}$ uniquely determines $R^T$ for any $T&gt;0$. A procedure recovering $R^\infty $ via $R^{2T_0}$ is also described.
LA - eng
KW - isotropic elasticity; dynamical Lame system; regularity of solutions; structure of sets reachable from the boundary in a short time; boundary controllability
UR - http://eudml.org/doc/245575
ER -

References

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