Consensus of the Hegselmann–Krause opinion formation model with time delay

In this paper, we study Hegselmann–Krause models with a time‐variable time delay. Under appropriate assumptions, we show the exponential asymptotic consensus when the time delay satisfies a suitable smallness assumption. Our main strategies for this are based on Lyapunov functional approach and careful estimates on the trajectories. We then study the mean‐field limit from the many‐individual Hegselmann–Krause equation to the continuity‐type partial differential equation as the number N of individuals goes to infinity. For the limiting equation, we prove global‐in‐time existence and uniqueness of measure‐valued solutions. We also use the fact that constants appearing in the consensus estimates for the particle system are independent of N to extend the exponential consensus result to the continuum model. Finally, some numerical tests are illustrated.