Trajectory controllability of stochastic integrodifferential equation with state-dependent delay driven by mixed Brownian motion suffered by non-instantaneous impulses

We develop a comprehensive framework for the analysis of mild solutions to non-instantaneous impulsive stochastic integrodifferential equations (SIDEs) with state-dependent delays and mixed Brownian motion in Hilbert spaces. The existence and uniqueness of solutions are established through tools from stochastic analysis, fixed-point theory and resolvent operator techniques. In particular, we present three distinct sets of sufficient conditions that guarantee the solvability of the proposed stochastic system. Moreover, under suitable and physically reasonable assumptions, we investigate the trajectory (T-)controllability of the system by employing an extended version of Gronwall's inequality. To illustrate and validate the theoretical results, we provide representative examples supported by numerical simulations. In addition, the model is extended to the fractional-order setting, where the fractional derivative is considered in the sense of piecewise continuous derivatives. For the numerical implementation, we employ exponential decay kernels, generalised Mittag–Leffler processes, and power-law processes, which capture a broad class of memory and hereditary effects. This manuscript significantly extends and unifies several existing results reported in George [(1995). Approximate controllability of nonautonomous semilinear systems. Non-Linear Analysis: Theory, Methods & Applications, 24(9), 1377–1393], Anguraj and Ramkumar. [(2018). Approximate controllability of a semilinear stochastic integrodifferential system with nonlocal conditions. Fractal and Fractional, 2(4), 29], Balachandran et al. [(1995). Controllability of non-linear integrodifferential systems in Banach space. Journal of Optimization Theory and Applications, 84(1), 83–91], Chalishajar [(2008). Controllability of non-linear integro-differential third order dispersion system. Journal of Mathematical Analysis and Applications, 348(1), 480–486], Chalishajar et al. [(2010). Trajectory controllability of non-linear integro-differential system. Journal of the Franklin Institute, 347(7), 1065–1075], Chalishajar and Chalishajar [(2015). Trajectory controllability of second-order non-linear integro-differential system: An analytical and a numerical estimation. Differential Equations and Dynamical Systems, 23(4), 467–481], Muslim and George [(2019). Trajectory controllability of the non-linear systems governed by fractional differential equations. Differential Equations and Dynamical Systems, 27(4), 529–537], Durga et al. [(2022). Trajectory controllability of Hilfer fractional neutral stochastic differential equation with deviated argument and mixed fractional Brownian motion. Optimisation, 72(11), 2865–2891] and Dhayal et al. [(2021). Approximate and trajectory controllability of fractional stochastic differential equations with non-instantaneous impulses and Poisson jumps. Asian Journal of Control, 23(6), 2669–2680]. To the best of our knowledge, the present work offers a novel and integrated combination of rigorous theoretical analysis and systematic numerical investigation for impulsive stochastic systems with delays and fractional dynamics.