Approximate Controllability of Higher-Order Hilfer Fractional Neutral Stochastic Systems Driven by Fractional Brownian Motion, Poisson Jumps, and 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.

This work focuses on the existence and trajectory (T-)controllability of mixed fractional Brownian motion (fBm) with the Hurst index ( 1 2 , 1 ) and neutral stochastic integrodifferential equations (NSIDEs) with deviating argument and fBm. Stochastic integrodifferential equations (SIDEs) are solved in Hilbert space using stochastic analysis, the resolvent operator, and Krasnoselskii's fixed point theorem (KFPT). Furthermore, providing adequate assumptions, the T-controllability of the considered system is organised by using extended Gronwall's inequality. We demonstrate the theoretical insights and numerical simulations are included which is unique and makes this work more interesting. The obtained results generalise existing results from [Chalishajar, D. N., George, R. K., & Nandakumaran, A. K. (2010). Trajectory controllability of nonlinear integro-differential system. Journal of Franklin Institute, 347(7), 1065–1075.; Durga, N., Muthukumar, P., & Malik, M. (2022). Trajectory controllability of Hilfer fractional neutral stochastic differential equation with deviated argument and mixed fractional Brownian motion. Optimisation, 1–27.; Muslim, M., & George, R. K. (2019). Trajectory controllability of the nonlinear systems governed by fractional differential equations. Differential Equations and Dynamical Systems, 27, 529–537.; Dhayal, R., Malik, M., & Abbas, S. (2021). Approximate and trajectory controllability of fractional stochastic differential equation with non-instantaneous impulses and Poisson jumps. Asian Journal of Control, 23(6), 2669–2680.].

In this paper, we introduce the mild solution for a new class of noninstantaneous and nonlocal impulsive Hilfer fractional stochastic integrodifferential equations with fractional Brownian motion and Poisson jumps. The existence of the mild solution is derived for the considered system by using fractional calculus, stochastic analysis and Sadovskii’s fixed point theorem. Finally, an example is also given to show the applicability of our obtained theory.

The existence of solutions of non-instantaneous impulsive Hilfer–Katugampola fractional differential equations of order 1 / 2 < α < 1 and parameter 0 ≤ β ≤ 1 with fractional Brownian motion (fBm) and Poisson jumps is investigated in this paper. The required results are obtained based on fractional calculus, stochastic analysis, semigroups, and the fixed point theorem. In the end of the paper, an example is provided to illustrate the applicability of the theoretical results.

In this paper, we establish sufficient conditions for exact null controllability of Sobolev type stochastic differential equations with fractional Brownian motion and Poisson jumps in Hilbert spaces, where the time fractional derivative is the Hilfer derivative. The exact null controllability result is derived by using fractional calculus, compact semigroup, fixed point theorem and stochastic analysis. Finally, an example is given to show the application of our results.

The aim of this manuscript is to analyze the existence of mild solution of non-instantaneous impulsive Hilfer fractional stochastic differential equations (NIHFSDEs) driven by fractional Brownian motion (fBm). Sufficient conditions for a class of NIHFSDEs of order and of type driven by fBm is derived with the help of fractional calculus, stochastic theory, fixed point theorem and semigroup theory. Mönch fixed point theorem (FPT) is adopted to prove the existence of solution. In addition, a numerical example is provided to validate the theoretical result.

In this manuscript, we establish a class of nonlocal Sobolev-type Hilfer fractional stochastic differential equations driven by fractional Brownian motion, which is a special case of a self-similar process, Hermite processes with stationary increments with long-range dependence. The Hermite process of order 1 is fractional Brownian motion and of order 2 is the Rosenblatt process. By using fractional calculus and fixed point approach, sufficient conditions of exact null controllability for such fractional stochastic systems are established. The derived result in this manuscript is new in the sense that it generalizes many of the existing results in the literature, more precisely for fractional Brownian motion and Poisson jumps case of Sobolev-type Hilfer fractional stochastic settings. Finally, stochastic partial differential equations are provided to validate the applicability of the derived theoretical results.

The objective of this paper is to investigate the existence of mild solutions and optimal controls for a class of fractional neutral stochastic differential equations (NSDEs) driven by fractional Brownian motion and Poisson jumps in Hilbert spaces. First, we establish a new set of sufficient conditions for the existence of mild solutions of the aforementioned fractional systems by using the successive approximation approach. The results are formulated and proved by using the fractional calculus, solution operator, and stochastic analysis techniques. The existence of optimal control pairs of system governed by fractional NSDEs driven by fractional Brownian motion and Poisson jumps is also been presented. An example is provided to illustrate the theory.

The aim of this manuscript is to analyze the stability for fractional neutral stochastic integro-differential delay systems (FNSIDDSs) with impulses driven by Poisson jumps. Sufficient conditions are established for a class of FNSIDDSs with impulses driven by Poisson jumps with the help of fractional calculus, stochastic theory, semigroup theory, and Mönch fixed point theorem in an infinite-dimensional space followed by the existence of a mild solution. The main reason for using Mönch fixed point theorem is its relation to Hausdorff measure of noncompactness (HMNC) adopted to prove the relative compactness conditions. Finally, an example is demonstrated to illustrate the obtained theoretical results.

In this paper, we investigate the existence of mild solutions and the approximate controllability of a class of nonlinear fractional stochastic differential equations of order \(1<q\le 2\) with infinite delay and Poisson jumps which satisfies the nonlocal conditions in Hilbert space. The existence of mild solutions is proved by using Sadovskii’s fixed point theorem. Also the approximate controllability of the nonlinear fractional nonlocal stochastic differential equations of order \(1<q\le 2\) with infinite delay and Poisson jumps is checked by using Lebesgue dominated convergence theorem. Finally an example is included to illustrate the results.

Using fractional calculus, stochastic analysis theory, and fixed point theorems with the properties of analytic \(\alpha \)-resolvent operators, sufficient conditions for approximate controllability of Sobolev-type fractional stochastic integrodifferential equations with fractional Brownian motion and Poisson jumps are established. Finally, an example is given to illustrate the obtained results.

In this manuscript, we investigate the existence, uniqueness, and exponential stability of a delayed neutral impulsive stochastic integro-differential equation driven by fractional Brownian motion in a separable Hilbert space and Poisson jumps. The results are obtained, using the theory of resolvent operators, stochastic analysis, and a fixed-point technique. Lastly, an example is provided to show the validity of the obtained results.

This paper focuses on a new class of non-instantaneous impulsive stochastic differential equations generated by mixed fractional Brownian motion with poisson jump in real separable Hilbert space. A set of sufficient conditions are generated based on the stochastic analysis technique, analytic semigroup theory of linear operators, fractional power of operators, and fixed point theory to obtain existence and uniqueness results of mild solutions for the considered system. Furthermore, the asymptotic behaviour of the system is investigated. Finally, an example is proposed to validate the obtained results.

&lt;abstract&gt;&lt;p&gt;The existence of a mild solution for nonlinear Hilfer fractional stochastic differential equations of the Sobolev type with non-instantaneous impulse in Hilbert space is investigated in this study. For nonlinear Hilfer fractional stochastic differential equations of Sobolev type with non-instantaneous impulsive conditions, sufficient criteria for controllability are established. Finally, an illustration of the acquired results is shown.&lt;/p&gt;&lt;/abstract&gt;

The paper is motivated by the study of interesting models from economics and the natural sciences where the underlying randomness contains jumps. Stochastic differential equations with Poisson jumps have become very popular in modeling the phenomena arising in the field of financial mathematics, where the jump processes are widely used to describe the asset and commodity price dynamics. This paper addresses the issue of approximate controllability of impulsive fractional stochastic differential systems with infinite delay and Poisson jumps in Hilbert spaces under the assumption that the corresponding linear system is approximately controllable. The existence of mild solutions of the fractional dynamical system is proved by using the Banach contraction principle and Krasnoselskii’s fixed-point theorem. More precisely, sufficient conditions for the controllability results are established by using fractional calculations, sectorial operator theory and stochastic analysis techniques. Finally, examples are provided to illustrate the applications of the main results.