Optimal controls for multi-term fractional stochastic integro-differential equations with impulses and infinite delay

This paper aims to investigate a new class of functional fractional stochastic impulsive integro-differential equations with infinite delay in Hilbert space with two Riemann–Liouville fractional derivatives. To demonstrate the existence of mild solutions for the considered systems, we use the fixed point approach, stochastic analysis, fractional calculus and $ (\alpha, \beta, \lambda ) $ (α, β, λ) resolvent family. Moreover, by utilising the Marzur lemma, we derived the optimal control results for the proposed problem. Finally, we provide an example to illustrate the obtained results.