Linear Matrix Inequality Approaches to Koopman Operator Approximation

The paper reformulates Koopman least-squares regression as a convex SDP with LMI constraints via slack variables Z,ν and the Schur complement, using scaled covariances G=(1/q)Θ+Ψ^T and H=(1/q)ΨΨ^T, exactly as in the candidate solution . It removes H^{-1} numerically via an eigendecomposition H=VΛV^T, yielding the inversion-free LMI [Z UV√Λ; √ΛV^T U^T I]≽0, again matching the candidate . The Tikhonov/spectral-norm regularization is encoded by the standard epigraph LMI [γI U^T; U γI]≽0 with objective term (α/q)γ^2, as in the model . Nuclear-norm regularization is implemented via the SDP [W1 U; U^T W2]≽0 and tr(W1)+tr(W2)≤2γ to enforce ∥U∥_*≤γ, again identical to the model’s construction . The asymptotic stability constraint uses a Lyapunov inequality together with a dilated LMI with a slack Γ and an alternating scheme over (A,P) and Γ, precisely as the model states (note the paper’s LMI carries ρ̄^2 in the top-left block) . Finally, H∞ regularization leverages the discrete-time bounded-real lemma with bilinearity handled by alternating between P and (A,B), also mirroring the model . The only discrepancy is a likely typographical omission of the square on ρ̄ in an intermediary inequality in the paper (A^T P A − ρ̄P ≤ 0), which is corrected in the LMI statement with ρ̄^2; the candidate consistently uses ρ̄^2. Overall, the approaches are the same and correct.