Problem 5.2
The dynamical Lame system with
boundary control:
on the structure of reachable sets
M. I. Belishev 1
Dept. of the Steklov Mathematical Institute (POMI)
Fontanka 27
St. Petersburg 191011
1 MOTIVATION
The questions posed below come from dynamical inverse problems for the hyperbolic systems with boundary control. These questions arise in the framework of the BC–method, which is an approach to inverse problems based on their relations to the boundary control theory [1], [2].
2 GEOMETRY
4 PROBLEMS AND HYPOTHESES
The open problem is to characterize the defect subspace 
. The following is the reasonable hypotheses.
• The defect space is always nontrivial: 
= for T < 
in the general case (not only in examples). Let us note that, due to the standard ‘controllability-observability’ duality,
this property would mean that in any inhomogeneous isotropic elastic media there exist the slow waves whose forward front propagates with the velocity 
.
• In the subdomain 
, where the elements of the defect subspace are supported, the pressure component of the wave ( see (*) ) determines its shear
component through a linear operator: 
in . If this holds, the question is to describe the operator 
.
• The decomposition (*) diagonalizes the principal part of the Lame system.
The progress in these questions would be of great importance for the inverse problems of the elasticity theory that is now the most difficult and challenging class of dynamical inverse problems.
BIBLIOGRAPHY
[1] M. I. Belishev, “Boundary control in reconstruction of manifolds and metrics (the BC-method), ” Inv.Prob., 13(5):R1–R45, 1997.
[2] M. I. Belishev and A. K. Glasman, “Dynamical inverse problem for the Maxwell system: Recovering the velocity in the regular zone (the BC-method), ” St.Petersburg Math. J., 12 (2):279–316, 2001.
[3] M. I. Belishev and I. Lasiecka, “The dynamical Lame system: Regularity of solutions, boundary controllability and boundary data continuation, ” ESAIM COCV, 8:143–167, 2002.
1Supported by the RFBR grant No. 02-01-00260.