We study McKean-Vlasov PDEs obained as the mean field limit of 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 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.