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

Abstract

We consider the problem of damping a control system with aftereffect described by first-order functional-differential equations on a temporal star graph. The delay in the system is proportional to time and propagates through the interior vertex. A variational problem of minimizing an energy functional is studied taking into consideration the probabilities of scenarios corresponding to different edges. It is established that the optimal trajectory satisfies Kirchhoff type conditions at the interior 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.