Finite-approximate Controllability for Partial Neutral Functional Evolution Equations

In this article, we introduce the concepts of finite-approximate controllability in the framework of semilinear neutral functional differential equations, focusing on first-order systems within a separable Hilbert space. We establish sufficient conditions for achieving finite-approximate controllability of the semilinear neutral functional differential equation by demonstrating that if the linear part of the system is approximately controllable, then under suitable conditions, the nonlinear part is also finite-approximately controllable. Our approach relies on fixed-point techniques to derive these results. Finally, we provide an example to show the applicability of our theoretical results.