The dynamical Lame system : regularity of solutions, boundary controllability and boundary data continuation
ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002) , pp. 143-167.

The boundary control problem for the dynamical Lame system (isotropic elasticity model) is considered. The continuity of the “input $\to$ 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 $u\left(·,T\right)$ of the complete state $\left\{u\left(·,T\right),{u}_{t}\left(·,T\right)\right\}$ 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 $\to$ output” map) ${R}^{2{T}_{0}}$ uniquely determines ${R}^{T}$ for any $T>0$. A procedure recovering ${R}^{\infty }$ via ${R}^{2{T}_{0}}$ is also described.

DOI : https://doi.org/10.1051/cocv:2002058
Classification : 93C20,  74B05,  35B65,  34K35
Mots clés : isotropic elasticity, dynamical Lame system, regularity of solutions, structure of sets reachable from the boundary in a short time, boundary controllability
@article{COCV_2002__8__143_0,
author = {Belishev, M. I. and Lasiecka, Irena},
title = {The dynamical Lame system : regularity of solutions, boundary controllability and boundary data continuation},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {143--167},
publisher = {EDP-Sciences},
volume = {8},
year = {2002},
doi = {10.1051/cocv:2002058},
zbl = {1064.93019},
mrnumber = {1932948},
language = {en},
url = {www.numdam.org/item/COCV_2002__8__143_0/}
}
Belishev, M. I.; Lasiecka, I. The dynamical Lame system : regularity of solutions, boundary controllability and boundary data continuation. ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002) , pp. 143-167. doi : 10.1051/cocv:2002058. http://www.numdam.org/item/COCV_2002__8__143_0/

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