Study on Control Problem of a Impulsive Neutral Integro-Differential Equations with Fading Memory

The paper builds an impulsive resolvent framework, defines the linear input–output map M and its adjoint M*, and expresses the Gramian MM* explicitly, then selects feedback controls via M* applied to a resolvent of the duality mapping. Under (H2), it invokes a result (from [5]) to obtain compactness of the resolvent family R(t) for t>0 and proves the fixed-point map Gα is compact by equicontinuity and pointwise relative compactness, enabling Schauder to yield a mild solution (Theorem 3.1). It then derives the terminal identity xα(b)=h−α(αI+MM*J)−1σ(x) and, using (H1) plus a Dunford–Pettis subsequence argument, shows ||xα(b)−h||→0 (Theorem 3.2). These steps align closely with the model’s argument: the model defines the same feedback via M*J[(αI+MM*J)−1(h−z(x))], uses (H2) to get compactness of R(t) and complete continuity of the solution operator, applies Schauder to obtain xα, establishes the same terminal resolvent identity, and invokes (H1) to conclude approximate controllability. The paper’s proofs provide the missing compactness and compactness-based Arzelà–Ascoli details for Gα, while the model streamlines the final convergence using uniformity over a compact set of free terminal terms. In content and structure the proofs are substantially the same, differing only in minor technical presentation choices. Key correspondences appear in the mild solution formula (2.4), the M/M* and Gramian definitions, the compactness/Arzelà–Ascoli argument for Gα, the control design (3.9), and the terminal identity (3.19), culminating in Theorem 3.2 on approximate controllability .