  Consensus of autonomous agents is a benchmark problem in multi-agent control.
In this paper, we consider continuous-time averaging consensus policies (or
Laplacian flows) and their discrete-time counterparts over time-varying graphs
in presence of unknown but bounded communication delays. It is known that
consensus is established (no matter how large the delays are) if the graph is
periodically, or uniformly quasi-strongly connected (UQSC). The UQSC condition
is often believed to be the weakest sufficient condition under which consensus
can be proved. We show that the UQSC condition can actually be substantially
relaxed and replaced by a condition that we call aperiodic quasi-strong
connectivity (AQSC), which, in some sense, proves to be very close to the
necessary condition of integral connectivity. Furthermore, in some special
situations such as undirected or type-symmetric graph, we find a necessary and
sufficient condition for consensus in presence of bounded delay; the relevant
results have been previously proved only in the undelayed case. The consensus
criteria established in this paper generalize a number of results known in the
literature.
