Control Strategies for the McKean-Vlasov Equation

Abstract

We study the non-linear McKean-Vlasov equation ∂_tμ = σΔμ + ∇ · [μ(∇V + ∇W ∗ μ)] , μ(x, 0) = μ₀(x), on a domain Ω ⊂ R^d with periodic boundary conditions, where σ > 0 is the diffusion parameter, V is a confining potential, and W is an interaction potential. This equation arises as the mean-field limit of the interacting particle system dxⁱ_t = −∇V(xⁱ_t) dt − (1/N) Σ^N_j=1 ∇W(xⁱ_t − x^j_t) dt + (2σ)^1/2 dBⁱ_t, as N → ∞ (see, e.g., [5]). Applications include opinion dynamics, synchronization [3], and collective motion [2]. Under the convexity of W and the uniform convexity of V, the existence and uniqueness of a stationary state μ̄ and exponential convergence (via a logarithmic Sobolev inequality [4]) are well established. Even under convexity, a small spectral gap can lead to slow convergence, motivating acceleration. In the absence of convexity, multiple steady states may arise, and one may wish to guide the dynamics toward a particular equilibrium. To accelerate convergence toward a desired steady state μ̄, we introduce control functions V(x) → V(x) + Σ^m_j=1 α_j(x)u_j(t), where α_j(x) are spatial shape functions and u_j(t) are time-dependent control inputs. Our objective is to design α_j and u_j so that the distribution μ converges to μ̄ as fast as possible. Setting y = μ − μ̄, ẏ = Ay + W(y + μ̄) − W(μ̄) + Σ^m_j=1 u_j(t)N_jy + Σ^m_j=1 u_j(t)N_jμ̄ is the controlled McKean-Vlasov equation in abstract form with Aμ := ∇ · (μ∇V + σ∇μ), N_jμ := ∇ · (μ∇α_j), and W(μ) := ∇ · (μ∇W ∗ μ). Following the approach in [1] for the (linear) Fokker-Planck equation, we linearize the nonlinear operator W about μ̄ via its Fréchet derivative, i.e., DW(μ̄)[y] ≈ W(y + μ̄) − W(μ̄). We then consider the quadratic cost functional J(u) = (1/2) ∫₀^∞ (〈y, My〉_L²(Ω) + ||u(t)||²₂) dt. By solving the associated Riccati equation for the linearized system, we derive an optimal feedback law that drives the solution of (1) to μ̄. The spatial control functions α_j are fixed by selecting them from the Fourier basis. [...]