  To achieve control objectives for extremely large-scale complex networks
using standard methods is essentially intractable. In this work a theory of the
approximate control of complex network systems is proposed and developed by the
use of graphon theory and the theory of infinite dimensional systems. First,
graphon dynamical system models are formulated in an appropriate infinite
dimensional space in order to represent arbitrary-size networks of linear
dynamical systems, and to define the convergence of sequences of network
systems with limits in the space. Exact controllability and approximate
controllability of graphon dynamical systems are then investigated. Second, the
minimum energy state-to-state control problem and the linear quadratic
regulator problem for systems on complex networks are considered. The control
problem for graphon limit systems is solved in each case and approximations are
defined which yield control laws for the original control problems.
Furthermore, convergence properties of the approximation schemes are
established. A systematic control design methodology is developed within this
framework. Finally, numerical examples of networks with randomly sampled
weightings are presented to illustrate the effectiveness of the graphon control
methodology.
