Graphon Control of Large-scale Networks of Linear Systems

Abstract

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.

arXiv comment16 pages. To appear in the IEEE Transactions on Automatic Control, 2020. Originally announced July 2018

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• Continuum limits of coupled oscillator networks depending on multiple sparse graphs
• Subspace Decomposition for Graphon LQR: Applications to VLSNs of Harmonic Oscillators
• Opinion Dynamics on Graphon: the piecewise constant case
• Capturing the critical coupling of large random Kuramoto networks with graphons
• Persistence of steady-states for dynamical systems on large networks