Commanded Fission and Fusion of Flocks: Weighted Cheeger Certificates for Organized Disaggregation, Tracking, and Re-Aggregation

Abstract

Many missions require a flock to split apart on command into prescribed sub-groups, hold those sub-groups as coherent flocks while each tracks its own objective, and then re-aggregate on command. Existing fission–fusion controllers are largely bio-inspired heuristics without convergence guarantees. We give a certificate-based framework in which each phase of the maneuver is governed by a weighted isoperimetric (Cheeger) inequality. The organizing observation is that organized disaggregation is the fragmentation criterion of flocking used as an objective rather than avoided as a hazard: the same Fiedlergradient that a connectivity-maintenance controller ascends to keep a flock together is descended, block-wise, to pull it apart along a commanded cut. We prove (i) a two-way fission certificate bounding the global consensus rate by the cross-cut conductance while each sub-flock retains a Neumann-certified cohesion rate; (ii) a multiway fission certificate a higher-order Cheeger bound μk−1 2 maxj ϕ(Vj) that certifies commanded k-way splitting and yields a spectral-gap detector for completion; (iii) a per-group Dirichlet tracking certificate for the hold phase; and (iv) a Neumann re-cohesion certificate for commanded fusion. The resulting three-mode switched controller is analyzed with a dwell-time argument, and we address the eigenvalue-crossing at the split instant by handing off from the spectral surrogate to the conductance gradient, which stays single-valued. A simulated disaggregate–hold–reaggregate cycle confirms every certificate: the global algebraic connectivity collapses to zero under fission while each block stays above threshold, the higher-order spectral gap opens, and connectivity is restored on the fusion command.