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.

翻译：我们研究由噪声驱动的相互作用粒子系统，用于模拟诸如观点动力学等现象。我们关注在平均场极限下表现出相变（即对应McKean-Vlasov PDE稳态解非唯一性）的系统。基于谱伽辽金近似并结合处理解多重性的紧缩牛顿法，我们开发了一种高效数值方案，用于识别平均场McKean-Vlasov PDE的所有稳态（包括稳定与不稳定态）。在找到所有可能的平衡态后，我们构建了一种最优控制策略，用于将系统动力学引导至选定的不稳定稳态。该控制通过滚动时域框架下的迭代开环求解器进行计算。我们在多个示例中验证了所提出的稳态计算与稳定化方法的有效性，包括观点动力学的噪声Hegselmann-Krause模型和生物物理学的Haken-Kelso-Bunz模型。数值实验证实了该方法能够捕捉这些系统丰富的自组织景观，并稳定目标不稳定构型。所提出的计算框架为理解和控制噪声驱动相互作用粒子系统的集体行为开辟了新途径，在社交动力学、生物同步及物理与社会系统中的集体行为等多个领域具有潜在应用价值。