On damping a control system on a star graph with global time-proportional delay

We consider the problem of damping a control system with delay, described by first-order functional-differential equations on a temporal star graph. The delay in the system is time-proportional and propagates through the internal vertex. We study the variational problem of minimizing the energy functional, taking into account the probabilities the of scenarios corresponding to different edges. It is established that the optimal trajectory satisfies Kirchhoff-type conditions at the internal vertex. The equivalence of the variational problem to a certain boundary value problem for second-order functional-differential equations on the graph is proved, and the unique solvability of both problems is established.