Dynamical inverse problem for two-velocity systems on finite trees

We consider the dynamical inverse problem for two velocity systems on finite trees in a time-optimal setting: i.e. we assume that the dynamical Dirichlet-to-Neumann map, which we use as inverse data, is known on some finite interval (the length of this interval depends on the optical diameter of a tree). Using the controllability of a dynamical system and ideas of the Boundary Control method, we can extract the spectral data from the dynamical one, and then extend the dynamical inverse data by an explicit formula, provided we understand it in a suitable (generalized) sense. Then we can construct the Titchmarsh-Weyl function and solve the inverse problem using the leaf-peeling method.