  The constrained consensus problem considered in this paper, denoted interval
consensus, is characterized by the fact that each agent can impose a lower and
upper bound on the achievable consensus value. Such constraints can be encoded
in the consensus dynamics by saturating the values that an agent transmits to
its neighboring nodes. We show in the paper that when the intersection of the
intervals imposed by the agents is nonempty, the resulting constrained
consensus problem must converge to a common value inside that intersection. In
our algorithm, convergence happens in a fully distributed manner, and without
need of sharing any information on the individual constraining intervals. When
the intersection of the intervals is an empty set, the intrinsic nonlinearity
of the network dynamics raises new challenges in understanding the node state
evolution. Using Brouwer fixed-point theorem we prove that in that case there
exists at least one equilibrium, and in fact the possible equilibria are
locally stable if the constraints are satisfied or dissatisfied at the same
time among all nodes. For graphs with sufficient sparsity it is further proven
that there is a unique equilibrium that is globally attractive if the
constraint intervals are pairwise disjoint.
