On damping a control system on a star graph with global time-proportional delay
A. P. Lednov
TL;DR
The paper addresses damping a control system with global, time-proportional delay on a temporal star graph. It adopts a variational approach, minimizing an edge-weighted energy functional across the star with a probabilistic interpretation of edge scenarios. The main contributions are (i) showing that the optimal trajectory satisfies Kirchhoff-type conditions at the internal vertex, and (ii) proving the equivalence of the variational problem with a boundary-value problem for second-order functional-differential equations on the graph, together with a proof of unique solvability for both problems. This work advances the analysis of control with global delays on temporal graphs and has potential applications to networked control systems with distributed delays, often modeled by pantograph-type equations on graphs.
Abstract
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.