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 (2010)

  • 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 L2-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 T0 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 RT 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 (2010): 143-167. <http://eudml.org/doc/90643>.

@article{Belishev2010,
abstract = { The boundary control problem for the dynamical Lame system (isotropic elasticity model) is considered. The continuity of the “input → state" map in L2-norms is established. A structure of the reachable sets for arbitrary T>0 is studied. In general case, only the first component $u(\cdot ,T)$ of the complete state $\\{ u(\cdot ,T),u_t(\cdot ,T)\\}$ 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 T0 exceeds the time needed for shear waves to fill the entire domain, then the response operator (“input → output" map) $R^\{2T_0\}$ uniquely determines RT for any T>0. A procedure recovering R∞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.; isotropic elasticity; structure of sets reachable from the boundary in a short time; boundary controllability},
language = {eng},
month = {3},
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/90643},
volume = {8},
year = {2010},
}

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
DA - 2010/3//
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 → state" map in L2-norms is established. A structure of the reachable sets for arbitrary T>0 is studied. In general case, only the first component $u(\cdot ,T)$ of the complete state $\{ u(\cdot ,T),u_t(\cdot ,T)\}$ 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 T0 exceeds the time needed for shear waves to fill the entire domain, then the response operator (“input → output" map) $R^{2T_0}$ uniquely determines RT for any T>0. A procedure recovering R∞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.; isotropic elasticity; structure of sets reachable from the boundary in a short time; boundary controllability
UR - http://eudml.org/doc/90643
ER -

References

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