Controllability transition and nonlocality in network control

TL;DR

This paper investigates the challenges in controlling large networks, revealing that numerical control often fails due to ill-conditioned controllability Gramian and that increasing control inputs beyond a critical threshold ensures successful control, highlighting a controllability transition.

Contribution

It uncovers the controllability transition phenomenon and links control trajectory nonlocality with control input requirements in network control.

Findings

Numerical control fails with ill-conditioned Gramian.

Control trajectories are generally nonlocal in phase space.

Success rate abruptly increases with more control inputs.

Abstract

A common goal in the control of a large network is to minimize the number of driver nodes or control inputs. Yet, the physical determination of control signals and the properties of the resulting control trajectories remain widely under-explored. Here we show that: (i) numerical control fails in practice even for linear systems if the controllability Gramian is ill-conditioned, which occurs frequently even when existing controllability criteria are satisfied unambiguously; (ii) the control trajectories are generally nonlocal in the phase space, and their lengths are strongly anti-correlated with the numerical success rate and number of control inputs; (iii) numerical success rate increases abruptly from zero to nearly one as the number of control inputs is increased, a transformation we term numerical controllability transition. This reveals a trade-off between nonlocality of the control trajectory in the phase space and nonlocality of the control inputs in the network itself. The failure of numerical control cannot be overcome in general by merely increasing numerical precision---successful control requires instead increasing the number of control inputs beyond the numerical controllability transition.