# Exponentially stable solution of mathematical model based on graph theory of agents dynamics on time scales

## TL;DR

This paper investigates leader-following consensus in agent dynamics on arbitrary time scales, allowing variable step sizes, and provides conditions for stability using Grönwall inequality, supported by illustrative examples.

## Contribution

It introduces a leader-following model on arbitrary time scales with variable step sizes and proves stability conditions using Grönwall inequality.

## Key findings

- Established conditions for leader-following consensus on arbitrary time scales
- Demonstrated stability results through illustrative examples
- Extended graph theory models to non-uniform time scales

## Abstract

In this paper an emergence of leader-following model based on graph theory on the arbitrary time scales is investigated. It means that the step size is not necessarily constant but it is a function of time. We propose and prove conditions ensuring a leader-following consensus for any time scales using Gr\"{o}nwall inequality. The presented results are illustrated by examples.