An optimal control for non-autonomous second-order stochastic differential equations with delayed arguments

Optimal control of non-autonomous second-order stochastic differential equations with delayed arguments is indispensable for managing systems exposed to uncertainty, time-dependent dynamics, and historical influences. These equations underpin a wide range of applications, including finance, engineering, and biology, where it's imperative to make informed decisions that mitigate risks or maximize returns while considering the inherent randomness, evolving conditions, and the impact of past states. By employing optimal control techniques, we can devise strategies that are resilient to uncertainty, adaptable to changing circumstances, and capable of accounting for the memory effects of previous events. This empowers us to optimize system performance, bolster stability, and attain desired objectives in intricate and dynamic environments. So, the goal of this article is to introduce a novel model of second-order perturbed stochastic differential equations incorporating non-local finite delay and deviated arguments in the setting of Hilbert spaces. Moreover, essential criteria are presented to examine the existence of a mild solution and evaluate the potential for approximate and optimal control of the proposed system. These results have been obtained by using evolution operators, fixed point techniques, random analytic methods, and compact semigroup theory. Further, to support the theoretical results, the optimal controllability of our model was studied by considering the Lagrange problem. Finally, the results were applied to discuss the approximate controllability of a partial differential equation. These models have the potential to advance the understanding and application of optimal control techniques for a wider range of complex systems.