Optimal control of collective electrotaxis in epithelial monolayers

The paper sets up and solves optimal-control surrogates for “constant velocity” (minimizing ∫ v̇² with a terminal velocity constraint, and then a windowed constant-velocity objective), deriving adjoint conditions and computing controls numerically. It reports that achieving an (approximately) constant velocity requires an unrealistically large initial electric field that then “rapidly decays,” and with amplitude bounds the terminal error improves as the bound grows. Those claims are well documented in their Sections 3.6–3.7 and figures (e.g., the statement about a very large initial field and a near-constant effective signal seff, and the smax study) . However, the paper does not present a simple analytical characterization of the exact-control problem “v(t)≡v⋆ on [0,T],” even though its state equations are linear (Eq. (8) with zero initial conditions, Eq. (9)) . The candidate solution gives that analytic necessity: v̇=0 forces seff≡(γ/α)v⋆ and, with the model’s İ=(s−I)/τa and ṡeff=(s−seff−I)/τe, one must have s(t)=seff+I(t)= (γ/α)v⋆(1+t/τa), i.e., a linearly increasing control; also, exact constancy on an interval containing t=0 is impossible without impulses. This derivation is correct for the paper’s linear dynamics and clarifies feasibility under amplitude bounds. In contrast, the paper’s narrative line that a very large initial field then “rapidly decays” to maintain a constant seff is not consistent with the linear state equations if v is exactly constant; exact constancy requires ṡ=seff/τa>0 thereafter. The paper’s numerical results address different costed objectives (budgeted or bounded control, terminal constraints, windowed constancy) and so do not contradict the candidate’s exact-control necessity; they are simply a different problem. Net: the model’s exact-control analysis is correct and fills a gap; the paper’s treatment is insightful but incomplete on this specific analytic point.