Exponential stability, T-controllability and optimal controllability of higher-order fractional neutral stochastic differential equation via integral contractor

The existence, uniqueness, exponential stability with the trajectory (T-)controllability and optimal control results for mild solutions to the fractional neutral stochastic differential system (FNSDSs) are presented in this article. To demonstrate the results, the concept of bounded integral contractors combined with the regularity, a weaker notion of Lipschitz continuity, with stochastic approach and sequencing techniques are used. In contrast to previous publications, we do not need to specify the induced inverse of the controllability operator to prove the stability results, and the relevant nonlinear function does not have to meet the Lipschitz condition. Furthermore, exponential stability result for FNSDSs with Poisson jump via impulsive integral inequality is established following the trajectory (T-) controllability for higher-order FNSDSs via integral contractors with the help of Gronwall’s inequality and the optimal control problem for higher-order FNSDSs via Balder’s theorem. A numerical example is discussed to justify the theory. Finally, a filtration model and the real life stochastic Kelvin–Voigt and Maxwell models with the numerical simulation are demonstrated to satisfy the acquired results. This paper extends all previous works having the nonlinear Lipschitz continuous operators.