Solvability and optimal control for second-order stochastic differential systems under the influence of delay and impulses

We explore a new kind of second-order impulsive stochastic differential system with delays. By using the Hausdorff measure of noncompactness and Mo¨nch fixed point theorem, we investigated the existence of mild solution for the considered systems. Further, we proved the existence of optimal control of the proposed system by using minimizing sequence, Lagrange functional, and Marzur lemma. An example is provided to validate the obtained results.