Identification problem for the wave equation with Neumann data input and Dirichlet data observations

Xiaobing Feng, Suzanne Lenhart, Vladimir Protopopescu, Lizabeth Rachele, and Brian Sutton

Nonlinear Anal. 52 (2003), no. 7, 1777–1795.

We seek to identify the dispersive coefficient in a wave equation with Neumann boundary conditions in a bounded space-time domain from imprecise observations of the solution on the boundary of the spatial domain (Dirichlet data). The problem is regularized and solved by casting it into an optimal control setting. By letting the "cost of the control" tend to zero, we obtain the limit of the corresponding control sequence, which we identify with the sought dispersive coefficient. The corresponding solution of the wave equation is interpreted as the possibly nonunique projection of the observation vector onto the range of the Neumann-to-Dirichlet maps corresponding to a single input Neumann data, as the dispersive coefficient is varied. Several numerical examples illustrate the merits and limitations of the procedure.