Modern Koopman Theory for Dynamical Systems

The paper derives ż = Λz + Σ_j L_{h_j}T(x) u_j in Koopman–eigenfunction coordinates and states the system is bilinearizable when the control Lie actions lie in the span of the chosen eigenfunctions, citing prior work for details. Under the additional (standard) assumption that the span closure uses constant coefficients, this yields d/dt z = A z + Σ_j u_j B_j z with constant A,B_j, exactly as the model shows. Thus, both arguments align; the model makes the coefficient-constancy assumption explicit, while the paper references it implicitly via the bilinearizability condition and citations. See the paper’s eqs. (6.37)–(6.40) and discussion around bilinearization and the Lie-operator formulation with inputs .