This study is devoted to control problems governed by a second-order semilinear evolution equation with a memory kernel \(\kappa \in \textrm{L}^{1}(0,\infty )\) satisfying \(\int _{0}^{\infty }\kappa (t)dt<1\) and the function \(\varkappa (t):=\int _{t}^{\infty }\kappa (s)ds\) is of positive type. In this process, we first discuss the well-posedness of the linear evolution equation with memory by introducing the concept of a resolvent family. We then examine several fundamental properties of the resolvent family \(\mathscr {R}(\cdot )\) and the associated family \(\mathscr {P}(\cdot )\) that are crucial for the development of our main results. Subsequently, we study a linear–quadratic regulator problem in order to derive an optimal control leading to the approximate controllability of the state and its derivative for the linear control system. Furthermore, we determine sufficient conditions for the existence of a mild solution of the semilinear control system, corresponding to a given control \(u\in \textrm{L}^2(J;\mathbb {U})\) , by using the Schaefer fixed point theorem. We also investigate controllability of a semilinear system in a Hilbert space by constructing a single control that ensures the weak approximate controllability of both the state and its derivative. Our main results are determined by combining the fixed point techniques with the approximation solvability method and weak topology. Finally, the theoretical results are applied to analyze the weak approximate controllability problem for a wave equation with a weakly singular kernel of the form \(\kappa (t)=e^{-at}\frac{t^{\nu -1}}{\Gamma (\nu )}, \ a>1, \ 0<\nu <1 \) .
