Controllability for the wave equation on graph with cycle and delta-prime vertex conditions

Exact controllability for the wave equation on a metric graph con-sisting of a cycle and two attached edges is proven. One boundary and one internal control are used. At the internal vertices, delta-prime conditions are satisfied. As a second example, we examine a tripod controlled at the root and the junction, while the leaves are fixed. These examples are key to understand-ing controllability properties in general metric graphs.