Gradient-based parameter calibration of an anisotropic interaction model for pedestrian dynamics

The paper proves existence of a minimizer for the parameter-calibration problem by a direct-method argument: it extracts weakly convergent subsequences in Y × Uad, uses weak continuity of the state operator e, and invokes weak lower semicontinuity of J to conclude Theorem 2 (existence of (y*,u*)) , with the state well-posedness built on Assumption 1 and Theorem 1 (Picard–Lindelöf) . The candidate solution instead reduces the problem to minimizing the reduced cost j(u)=J(S(u),u) on the compact set Uad and argues existence via continuity of the control-to-state map S and Weierstrass. Under the paper’s regularity assumptions (global boundedness and local Lipschitz for K, and well-posedness of the ODE), the reduced functional approach is valid. Minor caveats: (i) the candidate’s continuity proof implicitly assumes a collision-free trajectory tube (uniform minimal separation), which is stronger than needed; one can obtain continuity on bounded sets directly from the assumed local Lipschitz/continuity of the vector field; (ii) the paper asserts that the specific force in (5) satisfies Assumption 1, which needs clarification near xi=xj due to the unit-direction factor; nevertheless, the main existence theorem only relies on Assumption 1–2 and remains correct when those assumptions hold .