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This manuscript is concerned with the trajectory controllability of Hilfer fractional neutral stochastic differential equation having deviated arguments and mixed fractional Brownian motion with the Hurst parameter. The proposed Hilfer fractional system's solvability in Hilbert space is acquired by employing fractional calculus, stochastic analysis, semigroup theory, and Krasnoselskii's fixed point theorem. Furthermore, under some suitable assumptions, the considered system's trajectory controllability is established using generalized Gronwall's inequality. Finally, an example is delivered to illustrate the obtained theoretical results.

Trajectory controllability of Hilfer fractional neutral stochastic differential equation with deviated argument and mixed fractional Brownian motion