  The paper naturally connects theory of quantum graphs, optimal control theory
and theory of functional-differential equations, and gives a new look at
quantum graphs as temporal networks. This means that the variable
parameterizing the edges is associated with time, while each internal vertex
opens several scenarios for the process flow. Under such settings, we extend
the problem of damping a first-order control system of the retarded type, which
was studied before only on an interval, to an arbitrary tree graph by employing
the recently suggested concept of the global delay. The latter means that the
delay imposed starting from the initial moment of time, associated with the
root of the tree, propagates through all its internal vertices. By minimizing
the energy functional, we arrive at the corresponding variational problem and
then prove its equivalence to a self-adjoint boundary value problem on the tree
for some second-order equations involving both the global delay and the global
advance, whose unique solvability is also established. Noteworthy is that at
the internal vertices, the optimal trajectory obeys Kirchhoff-type conditions,
which are common also for various models dealing with spacial networks.
