On the damping of a neutral-type control system on a temporal star graph with time-proportional delay

A. P. Lednov<sup>abc

a</sup> Saratov National Research State University named after N. G. Chernyshevsky, Saratov, Russia
^b Moscow Center of Fundamental and Applied Mathematics, Moscow, Russia
^c Lomonosov Moscow State University, Moscow, Russia

Abstract: On a temporal star graph, we consider the problem of optimally damping a control system for a generalized pantograph equation, which is a neutral-type equation with time-proportional delay. The delay in the system propagates through an internal vertex of the graph. We study the variational problem of minimizing the energy functional, taking into account the probabilities of scenarios corresponding to different edges. We establish that the optimal trajectory satisfies Kirchhoff-type conditions at the internal vertex. We prove the equivalence of the variational problem to a certain boundary-value problem for second-order functional differential equations on the graph, and establish the unique solvability of both problems.

Keywords: neutral-type equation with delay, pantograph equation, star graph, optimal system damping, Krasovskii problem, variational problem, unique solvability.

UDC: 517.9

DOI: 10.22363/2413-3639-2025-71-4-642-654