Controllability of networks: influence of structure and memory

In this work we borrow some ideas from the the theory of lifted Markov chains, which can accelerate convergence of random walks algorithms thanks to the introduced memory effects, and apply them to the control of networks of dynamical systems arranged on a line and on a grid. We lift the dynamics by enlarging each node state and discuss how to compare the effect of a control input on the lifted network and on the original one. We compute some metrics for energy-related controllability, showing that the lifted network has better controllability properties than the non-lifted one. This proves an advantage induced by the extra internal dynamics that allows for memory effects. The potential of lifts is then explored via numerical simulations for some paradigmatic examples.