Jammed granular systems are nonlinear. Forces propagate through irregular contact networks. Rearrangements are abrupt, discontinuous, and sensitive to details of the packing. The standard tools for analyzing them — force chains, coordination number, vibrational density of states — are built to handle this nonlinearity. Nobody reaches for linear control theory, the mathematics of systems that respond proportionally to inputs, because nothing about a jammed packing responds proportionally.
Teich, Kim, and Bassett applied linear control theory anyway. They computed the average controllability of each particle — a measure from network control theory that quantifies how much a node can influence the system's state through the linearized dynamics. Then they sheared the packing and watched which particles rearranged.
Average controllability predicts rearrangement. The linear measure, computed from the system's vibrational eigenmodes, identifies which particles will move under quasistatic shear — a deeply nonlinear process. As shear increases toward a rearrangement event, the participating particles progressively engage lower-energy eigenmodes. The linear framework captures the approach to instability because the instability itself is a loss of linear stability — the nonlinear event is preceded by a linear signature.
The result positions control theory not as an approximation that fails at the interesting parts but as a tool that sees the interesting parts coming. The rearrangement is nonlinear. The warning is linear. The mismatch between the tool and the system is exactly what makes the tool useful — it detects the regime where the system is about to leave the linear neighborhood, which is precisely when rearrangement happens.