  The linear-quadratic controller is one of the fundamental problems in control
theory. The optimal solution is a linear controller that requires access to the
state of the entire system at any given time. When considering a network
system, this renders the optimal controller a centralized one. The
interconnected nature of a network system often demands a distributed
controller, where different components of the system are controlled based only
on local information. Unlike the classical centralized case, obtaining the
optimal distributed controller is usually an intractable problem. Thus, we
adopt a graph neural network (GNN) as a parametrization of distributed
controllers. GNNs are naturally local and have distributed architectures,
making them well suited for learning nonlinear distributed controllers. By
casting the linear-quadratic problem as a self-supervised learning problem, we
are able to find the best GNN-based distributed controller. We also derive
sufficient conditions for the resulting closed-loop system to be stable. We run
extensive simulations to study the performance of GNN-based distributed
controllers and showcase that they are a computationally efficient
parametrization with scalability and transferability capabilities.
