Finding similarity of orbits between two discrete dynamical systems via optimal principle

The paper obtains the stationarity condition (3.16) for the simple case by writing J(A)=∑||Ax_k−y_k||^2, expanding ∂J/∂a_{ij} termwise as in (3.6) and combining the componentwise derivatives (3.7), (3.11)–(3.15) to reach Proposition 3.1 (equation (3.16)) . For the homotopy case, it analogously sets up J(A,λ) as in (3.22) and applies KKT partial derivatives (3.23)–(3.24) to derive (3.31)–(3.32) . The candidate solution reproduces these conditions in a compact vector-Jacobian notation (DΦ_k), with the same chain-rule content as the paper’s expanded index formulas, and correctly explains the absence of ∂A/∂λ in (3.32) by noting that the derivative is a partial (holding A fixed) and, equivalently, by the envelope theorem at a joint stationary point—exactly how the paper treats (3.24) (no ∂A/∂λ terms) . Minor differences are expository (notation, explicit KKT/boundary comments), not mathematical.