Controllability of partial differential equations on graphs

We study the boundary control problems for the wave, heat, and Schr\"odinger equations on a finite graph. We suppose that the graph is a tree (i.e., it does not contain cycles), and on each edge an equation is defined. The control is acting through the Dirichlet condition applied to all or all but one boundary vertices. The exact controllability in $L_2$-classes of controls is proved and sharp estimates of the time of controllability are obtained for the wave equation. The null controllability for the heat equation and exact controllability for the Schr\"odinger equation in arbitrary time interval are obtained.