Approximate and optimal controllability of a second order non-autonomous stochastic differential equation with deviated arguments

This paper studies a second-order non-autonomous stochastic differential equation with deviated arguments and nonlocal finite delay in a Hilbert space. The objective is to provide sufficient conditions existence and uniqueness of the mild solution and the approximate and optimal controllability of the stochastic control system. We used Krasnoselskii's fixed point theorem, the theory of compact semigroup, evolution operators, and stochastic analysis techniques to establish these results. An example is included as an application to demonstrate the main result.