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

Belishev, M. I.; Lasiecka, I.

doi:10.1051/cocv:2002058

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

Belishev, M. I. ; Lasiecka, I.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 143-167

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Résumé

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 $u (\cdot, T)$ 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 $T_{0}$ 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.

MR Zbl

DOI : 10.1051/cocv:2002058

Classification : 93C20, 74B05, 35B65, 34K35
Keywords: 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, I.},
     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},
     mrnumber = {1932948},
     zbl = {1064.93019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv:2002058/}
}

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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

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