Controllability problem of an evolution equation with singular memory

This work addresses control problems governed by a semilinear evolution equation with singular memory kernel $\kappa(t)=\alpha e^{-\beta t}\frac{t^{\nu-1}}{\Gamma(\nu)}$, where $\alpha>0, \beta\ge 0$, and $0<\nu<1$. We examine the existence of a mild solution and the approximate controllability of both linear and semilinear control systems. To this end, we introduce the concept of a resolvent family associated with the linear evolution equation with memory and develop some of its essential properties. Subsequently, we consider a linear-quadratic regulator problem to determine the optimal control that yields approximate controllability for the linear control system. Furthermore, we derive sufficient conditions for the existence of a mild solution and the approximate controllability of a semilinear system in a super-reflexive Banach space. Additionally, we present an approximate controllability result within the framework of a general Banach space. Finally, we apply our theoretical findings to investigate the approximate controllability of the heat equation with singular memory.