Geodesic
On boundary controllability of dynamical system gouverned by the wave equation on a~class of graphs (trees)
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 33, Tome 308 (2004), pp. 23-47

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The boundary control problem for the wave equation on a planar graph consisting of strings of variable densities with Dirichlet control at the boundary vertices is considered. The exact controllability in $L_2$–classes of controls and states is established in the case that the grapf is a tree; a sharp estimate of the time of controllability is given. 
@article{ZNSL_2004_308_a2,
     author = {M. I. Belishev},
     title = {On boundary controllability of dynamical system gouverned by the wave equation on a~class of graphs (trees)},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {23--47},
     publisher = {mathdoc},
     volume = {308},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2004_308_a2/}
}
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M. I. Belishev. On boundary controllability of dynamical system gouverned by the wave equation on a~class of graphs (trees). Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 33, Tome 308 (2004), pp. 23-47. http://geodesic.mathdoc.fr/item/ZNSL_2004_308_a2/