Computation and control of unstable steady states for mean field multiagent systems

We study McKean–Vlasov PDEs obtained as the mean field limit of interacting particle systems driven by noise, modelling phenomena such as opinion dynamics. We are interested in systems that exhibit phase transitions, i.e. non-uniqueness of stationary states for the corresponding 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 equilibria, 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 behaviour of noise-driven interacting particle systems, with potential applications in various fields such as social dynamics, biological synchronization and collective behaviour in physical and social systems.