Computation and Control of Unstable Steady States for Mean Field Multiagent Systems
<|reference_start|>Computation and Control of Unstable Steady States for Mean Field Multiagent Systems: We study interacting particle systems driven by noise, modeling phenomena such as opinion dynamics. We are interested in systems that exhibit phase transitions i.e. non-uniqueness of stationary states for the corresponding McKean-Vlasov PDE, in the mean field limit. We develop an efficient numerical scheme for identifying all steady states (both stable and unstable) of the mean field McKean-Vlasov PDE, based on a spectral Galerkin approximation combined with a deflated Newton's method to handle the multiplicity of solutions. Having found all possible equilibra, we formulate an optimal control strategy for steering the dynamics towards a chosen unstable steady state. The control is computed using iterated open-loop solvers in a receding horizon fashion. We demonstrate the effectiveness of the proposed steady state computation and stabilization methodology on several examples, including the noisy Hegselmann-Krause model for opinion dynamics and the Haken-Kelso-Bunz model from biophysics. The numerical experiments validate the ability of the approach to capture the rich self-organization landscape of these systems and to stabilize unstable configurations of interest. The proposed computational framework opens up new possibilities for understanding and controlling the collective behavior of noise-driven interacting particle systems, with potential applications in various fields such as social dynamics, biological synchronization, and collective behavior in physical and social systems.<|reference_end|>
@article{bicego2024computation,
    title={Computation and Control of Unstable Steady States for Mean Field
  Multiagent Systems},
    author={Sara Bicego, Dante Kalise, Grigorios A. Pavliotis},
    journal={arXiv preprint arXiv:2406.11725},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.11725},
    primaryClass={math.OC cs.NA math.NA}
}