Well-posedness and properties of the flow for semilinear boundary control systems

The paper’s Theorem 3.7 establishes a uniform short-time existence/uniqueness result for mild solutions of ẋ = Ax + B2 f(x,u) + Bu under Assumptions 3.1–3.3 and a growth bound on T, with the radius estimate sup_{t≤t1} ||x(t)−w|| ≤ Mr + h0 ||u||∞ + δ; its proof uses a Picard map on C([0,t],X), admissibility bounds for B and B2 (h_t, c_t), and a contraction argument, followed by a compactness argument to make t1 uniform over w ∈ Q . The candidate model’s solution replicates this structure almost verbatim: it defines the same Picard operator Γ on C([0,t1],X), uses zero-class ∞-admissibility of B2 to get a small-time gain k_t → 0 and ∞-admissibility of B to get h_t with limit h0, proves Γ maps a suitable invariant ball into itself, and shows Γ is a contraction for small t1, then applies Banach’s fixed-point theorem and a continuation uniqueness argument. Minor differences are purely notational (e.g., using k_t instead of c_t and an innocuous bound R(u) ≤ R* + δ where R(u) ≤ R* suffices) and do not affect correctness. Note: the paper text contains an occasional T−1 in the B2-integral lines of the proof sketch, but the formal mild solution and the rest of the argument use T for B2 as intended; this appears to be a typographical slip, not a substantive issue .