(untitled) (European Journal of Pure and Applied Mathematics, 2025)
This study addresses the challenge of achieving scaled consensus in multi-agent systems with edge dynamics, a critical aspect for practical applications such as traffic networks and distributed energy systems. Utilizing graph theory, matrix analysis, and Lyapunov stability, we propose impulsive consensus protocols designed for both directed and undirected topologies. These protocols enable agents to converge proportionally to desired ratios among states, generalizing classical consensus approaches while significantly reducing communication and control costs. We establish sufficient conditions for achieving scaled consensus, demonstrating that it requires the line graph of the communication network to be connected and contain a spanning tree. Numerical simulations validate the robustness and efficiency of the proposed protocols, highlighting their scalability and adaptability in dynamic and resource-constrained environments. The findings provide a solid foundation for real-world applications, including autonomous vehicles, distributed energy systems, and robotic coordination, where proportional state alignment is essential.