  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.
