Source-linked AI summary
Controllability transition and nonlocality in network control
Jie Sun, Adilson E. Motter
TL;DR
The paper asks whether theoretically valid network-control signals can actually be computed, and examines controllability using Gramian conditioning and control-trajectory locality. It finds that ill-conditioned Gramians cause practical failure despite formal controllability, while increasing control inputs produces a sharp numerical controllability transition that precision alone generally cannot overcome.
Problem
The paper investigates whether control signals can be constructed in practice despite formal equivalence between Gramian invertibility and the Kalman rank condition.
Method
The paper analyzes linear network control through controllability-Gramian conditioning, minimum-energy trajectories, strict local controllability, and numerical experiments on network ensembles.
Results
Ill-conditioned controllability Gramians cause numerical control failure even when the Kalman controllability matrix is well-conditioned, while success rises sharply from zero to one as driver nodes increase.
Takeaways & Limitations
Practical controllability requires accounting for numerical Gramian rank and balancing nonlocal phase-space trajectories against nonlocal control inputs.
Takeaways & Limitations
Even deterministic, autonomous, linear systems exhibit a fundamental disparity between theoretical controllability and practical control of large networks.
Abstract
from arXiv · showhide
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.