  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.
