Controllability of suspension bridge model proposed by Lazer and Mckenna under the influence of impulses, delays, and non-local conditions

For approximate controllability, the paper proves the result by a short-time steering argument that replaces the control on [T−ς,T] by the exact linear steering control ũ derived from Theorem 2.2, and then shows the terminal nonlinear remainder can be made arbitrarily small under the growth condition (13) (see the construction of uς and ũ and the estimates around equations (14)–(16) and Theorem 3.1) . The candidate solution’s Part A instead uses the R(ε)-steering regularization plus a Schauder-type fixed-point argument, which is a different method that requires stronger hypotheses (approximate controllability of the linear pair and compactness of the semigroup) but still yields approximate controllability. For exact controllability with nonlinearities independent of u, the paper’s Theorem 4.1 defines the same feedback/affine control via Γ = B^*S^*(T−·)W^{-1} and proves that the associated fixed-point map is a contraction under ML_q + MT||B||||Γ||C + MT l + MN < 1; this proof is essentially identical to the candidate’s Part B (compare the definition of κ, L, and the contraction estimate leading to (18)) .