Recovering a speed of sound from partial data
We consider the problem of acoustical tomography: to find a speed of sound by partial boundary measurements using the Boundary Control method. In our statement Neumann sources are located at the same part of the boundary as receivers. The pressure at another part of the boundary is assumed to be zero. We prove H1 approximate boundary controllability and give an explicit linear algorithm of recovering a sound speed provided that the time of observation is big enough. This algorithm is based on solving Boundary Control problem (BCP) for harmonic functions (i.e. we are looking a boundary control which approximate in the Sobolev space H1 any prescribed harmonic function). First we prove some version of the Friedrichs-Poincaré inequality and obtain conditional estimate for BCP. Then the reconstruction of the sound speed is reduced to the inversion of Fourier transform.