## 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.

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