OPTIMAL CONTROL FOR A CLASS OF LINEAR TRANSPORT-DOMINATED SYSTEMS VIA THE SHIFTED PROPER ORTHOGONAL DECOMPOSITION

The paper’s FRTO system (33a–33e) and FOTR system (34a–34e) are internally consistent and follow from the detailed Lagrangian derivation with a state-dependent mass matrix, including the nontrivial E-block terms that collect derivatives of the mass and coupling matrices with respect to the shift-coordinates and amplitudes (cf. (22)–(23), (30)–(33)) . The candidate solution incorrectly asserts that the H′[ẋ] terms cancel in the FRTO derivation and consequently identifies the E-blocks with the Jacobian of g only; however, the paper’s E-blocks explicitly contain additional contributions involving M′, N′, D′, V′, W′, and the state time-derivatives ȧs, żs (see (30)) . The candidate also omits the subtle singularity at tf for the FOTR adjoint mass matrix when asa(tf)=0 (Remark 3.2) . The paper’s noncommutation statement in Remark 3.3 is correct and aligns with the presented systems (different adjoints, terminal conditions, and control–adjoint couplings) .