On damping neutral-type control system on a temporal star graph with global time-proportional delay
Alexsandr Lednov
TL;DR
Problem: damping a neutral-type pantograph system on a temporal star graph with global time-proportional delay ($q>1$). Approach: a weighted variational formulation over the edges with energy $\mathcal J$ and reduction to a boundary-value problem for second-order functional-differential equations on the graph, together with Kirchhoff-type vertex conditions; prove equivalence between the variational and boundary formulations and establish unique solvability under $|a_1|\neq q^{-1/2}$ and $\sum_{j=2}^m |a_j|>0$; discuss the neutral-only subcase. Key contributions: rigorous equivalence, coercive solvability, and explicit a priori estimates for the star-graph setting. Significance: provides a mathematically solid framework for optimal damping in time-delayed networked control systems and informs extensions to more complex graphs.
Abstract
We consider, on a temporal star graph, the problem of optimal damping a control system is considered for a generalized pantograph equation, which is a neutral-type equation with a time-proportional delay. The delay in the system propagates through the internal vertex of the graph. 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.