Control of collective dynamics with time-varying weights

The paper proves approximate controllability of the barycenter to any target in the interior of the initial convex hull under (i) mass-preserving controls u ∈ U_α^∞ ∩ U_M (Theorem 2.1) and (ii) mass-varying controls u ∈ U_α^∞ \ U_M (Theorem 3.1). For (i), it bounds agent motion via δ := sup_{s∈[0,D0]} s a(s) and designs a steepest-descent control (extreme agents i−, i+) to obtain an exponential decrease of ||x̄ − x*|| within a time window that keeps x* inside the shrinking hull . For (ii), it uses a constant control to drive weights to κτ^0 (with τ^0 strictly positive s.t. x* = Σ τ^0_i x^0_i) while keeping positions close, yielding the desired accuracy . The candidate solution proves the same end-statements with a different constructive method: it steers the weight fractions w_i=m_i/Σ m_k along a prescribed path (u_i = ẇ_i/w_i) in the mass-preserving case, and for the mass-varying case appends a short uniform “bump” u_i=c to exit U_M before applying the same w-steering. These arguments are sound and align with the system identities used in the paper (e.g., ˙x̄ = (1/S) Σ m_i u_i (x_i − x̄) in the non-conservative case, compare to Eq. (8) in the paper) . One technical caveat in the paper’s mass-preserving proof as excerpted: the two-component control choice ui− = α (m_{i+}/m_{i−}), ui+ = −α does not always satisfy |ui−| ≤ α when m_{i+} > m_{i−}; the fix is to use an extremal point with two or three active components of U_α^∞ ∩ U_M (as the paper itself notes later in its discussion), which preserves the conclusion. Overall, both are correct, with different proof styles.