New results about controllability and stability of Ψ-Caputo-type stochastic fractional integro-differential systems with control delay

Abstract This study explores the controllability and stability criteria for the fractional integro-differential stochastical systems with control delay employing the Ψ-Caputo-type fractional derivative (Ψ-CTFD) of order σ ∈ (0, 1). The necessary and sufficient conditions
for the controllability of linear stochastical systems are obtained by utilizing the positive definiteness of the Grammian matrix. The sufficient conditions for the controllability criteria of semilinear and integro-differential stochastical systems are derived by
employing Banach’s contraction principle. The sufficient conditions for the stability of integro-differential stochastical systems are derived from the Hyer’s-Ulam stability criteria. Additionally, a few examples and appropriate graphical representations are included
in the work to help understand the theoretical conclusions.