  We consider the problem of making a set of states invariant for a network of
controlled systems. We assume that the subsystems, initially uncoupled, must be
interconnected through controllers to be designed with a constraint on the data
rate obtained by every subsystem from all the other subsystems. We introduce
the notion of subsystem invariance entropy, which is a measure for the smallest
data rate arriving at a fixed subsystem, above which the overall system is able
to achieve the control goal. Moreover, we associate to a network of n
subsystems a closed convex set of R^n encompassing all possible combinations of
data rates within the network that guarantee the existence of corresponding
feedback strategies for making a given set invariant. The extremal points of
this convex set can be regarded as Pareto-optimal data rates for the control
problem, expressing a trade-off between the data rates required by different
systems. We characterize these quantities for linear systems, and for
synchronization of chaos.
