 
Summary: AN INVERSE INITIAL BOUNDARY VALUE PROBLEM FOR THE
WAVE EQUATION IN THE PRESENCE OF IMPERFECTIONS OF
SMALL VOLUME
HABIB AMMARI
SIAM J. CONTROL OPTIM. c 2002 Society for Industrial and Applied Mathematics
Vol. 41, No. 4, pp. 11941211
Dedicated to JeanClaude N´ed´elec for his 60th birthday
Abstract. We consider for the wave equation the inverse problem of identifying locations and
certain properties of the shapes of small conductivity inhomogeneities in a homogeneous background
medium from dynamic boundary measurements on part of the boundary and for a finite interval in
time. Using as weights particular background solutions constructed by a geometrical control method,
we develop an asymptotic method based on appropriate averaging of the partial dynamic boundary
measurements. Our approach is expected to lead to very effective computational identification algo
rithms.
Key words. inverse problem, wave equation, reconstruction, geometric control
AMS subject classifications. 35R30, 35B40, 35B37, 35L05
PII. S0363012901384247
1. The inverse initial boundary value problem. Let be a bounded,
smooth subdomain of R2
. For simplicity, we take to be C
