In this paper, we study a specific class of integro-differential systems within Hilbert spaces that aligns with the Coleman-Gurtin model of heat conduction incorporating memory effects. Motivated by the recent results in [1], we relax the regularity assumptions on the kernels compared to previous studies. The well-posedness of the system’s state equation is demonstrated through the application of semigroups and resolvents operator theories. By relying on Laplace transform methods, we derive sufficient conditions under which the finite-time (and infinite-time) admissibility of the system’s observation operator can be inferred from the corresponding finite-time admissibility of the same operator for the related first-order Cauchy system, which lacks convolution terms. The finite-time admissibility is established by integrating a perturbation semigroup approach with admissible observation operators, while the infinite-time admissibility is achieved using the semigroup method combining with the Hardy space technique. Finally, illustrative examples are presented.
