The dynamical Lame system: regularity of solutions, boundary controllability and boundary data continuation
Saint-Petersburg Department of
the Steklov Mathematical Institute (POMI), Fontanka 27,
St. Petersburg 191011, Russia; firstname.lastname@example.org.
2 Department of Mathematics, University of Virginia, Charlottesville, VA 22901, USA; email@example.com.
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 of the complete state 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) uniquely determines RT for any T>0. A procedure recovering R∞ via is also described.
Mathematics Subject Classification: 93C20 / 74B05 / 35B65 / 34K35
Key words: Isotropic elasticity / dynamical Lame system / regularity of solutions / structure of sets reachable from the boundary in a short time / boundary controllability.
© EDP Sciences, SMAI, 2002