Network entropy and data rates required for networked control

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.