Hamiltonian bridge: A physics-driven generative framework for targeted pattern control

The paper’s derivation of the Hamiltonian bridge optimal control (action with co-state, elimination of controls to obtain H, canonical field equations, and the dual max formulation) matches the candidate solution step-by-step. The optimal controls u* = −(1/γu)∇Λ and S* = −(1/γv)Λ, the Hamiltonian H(φ,λ), and the canonical PDEs ∂tφ = δH/δλ, ∂tλ = −δH/δφ appear identically (see S.15–S.18 and Eq. (4) for H, and Eq. (3) for the canonical flow ). Model A (Allen–Cahn: v≡0, u≡0) yields a time-constant optimal S and an explicit affine-in-time interpolant, exactly as in S3.1 (Eq. S.19) . Model B (Cahn–Hilliard: R≡0, v≡0, m(φ)=φ) gives the adjoint PDE ∂tΛ − (1/(2γu))||∇Λ||^2 = 0 and straight-line characteristics with constant velocity, as in S3.2 (Eq. (9)) . Adding intrinsic drift via H1 = −∫ φ ∇λ·∇U deforms paths with velocity −∇U − (1/γu) e^{∫∇^2U}∇Λ0 and acceleration ∇^2U∇U = −∇(−½||∇U||^2), exactly the paper’s Eq. (10) and discussion . Minor caveats (boundary/regularity assumptions, and a casual minimax interchange remark) do not affect the core equivalence.