A unifying framework for tangential interpolation of structured bilinear control systems

The paper’s Theorem 1 states right, left, and bi-tangential interpolation for the modified multivariate transfer functions under explicit subspace conditions on V and W, with K(s)−1 and K̂(s)−1 assumed defined at the used points. The proof in the paper is given in detail for Part (c) using Petrov–Galerkin oblique projectors and indicates Parts (a)–(b) follow analogously; in particular, the argument repeatedly inserts projectors P_v = V K̂(s)^{-1} W^H K(s) (and their left dual) to collapse chains and show V v̂_q = v_q and W ŵ_η = w_η, yielding c^H G_{q+η}(...) b = c^H Ĝ_{q+η}(...) b . The candidate solution explicitly introduces the same oblique projectors PR(s) and PL(s), proves the projector identities, and carries out detailed collapsing arguments to establish (a), (b), and then (c). This mirrors the paper’s method, with the extra identity u^H N(s|δ) V K̂(s)^{-1} W^H Y = (K(s)^{-H} N(s|δ)^H u)^H Y used to streamline the left-to-right peeling; this step is consistent with the paper’s recursive definition of left states and the subspace condition span(W) ⊇ span([w_1,...,w_κ]) ensuring u^+ ∈ range(W). Assumptions match the paper’s (K(s)−1, K̂(s)−1 exist) and the definition of N(s|d) as a scaled sum is standard in the paper’s framework . Hence, both are correct and use substantially the same projector-based proof.