Fractional Stochastic Differential System Research Articles

Stability analysis of damped fractional stochastic differential systems with Poisson jumps: an successive approximation approach

This paper aims to investigate a new class of fractional stochastic differential systems under the influence of damping and Poisson jumps. First, the existence and uniqueness of mild solutions for the proposed system are investigated under the non-Lipschitz conditions. The results are formulated and proved by using the ( β , δ ) -regularised family, Grönwall's inequality, and successive approximation technique, which is different from the fixed point approach. Moreover, novel stability criteria for the considered system are obtained by utilising the corollary of the Bihari inequality. Finally, the correctness of the obtained results is verified by example.

Existence results of a nonlocal impulsive fractional stochastic differential systems with Atangana–Baleanu derivative

Existence results of a nonlocal impulsive fractional stochastic differential systems with Atangana–Baleanu derivative

Approximate controllability for Hilfer fractional stochastic differential systems of order 1

In this paper, we analyse the approximate controllability of Hilfer fractional stochastic differential systems of order 1 < μ < 2 in Hilbert spaces. The primary findings are carried out by using fractional calculus, stochastic analysis theory, cosine family, and Schauder's fixed point theorem. In particular, we draw a new set of sufficient conditions for the approximate controllability of Hilfer fractional stochastic differential equations of order 1 < μ < 2 under the assumption that the corresponding linear system is approximately controllable. In the end, an example is provided to illustrate the derived theory.

Existence theory and stability analysis of neutral $ \psi $–Hilfer fractional stochastic differential system with fractional noises and non-instantaneous impulses

&lt;abstract&gt;&lt;p&gt;In this article, with the help of Laplace transform, the existence of solution was established in a finite dimensional setting for nonlinear $ \psi $-Hilfer fractional stochastic equation with both retarded and advanced arguments driven by multiplicative and fractional noises, with Hurst index $ H \in (\frac{1}{2}, 1) $. At first, we obtained the existence and uniqueness results by using the Banach fixed point theorem (FPT). Second, the existence result was also obtained by applying Schaefer's fixed point theorem with less conservative conditions. Furthermore, we investigated the Hyers Ulam Rasisas stability for the aforementioned system. At the end, an example was illustrated to validate the obtained theoretical results.&lt;/p&gt;&lt;/abstract&gt;

Existence and controllability of nonlinear evolution equation involving Hilfer fractional derivative with noise and impulsive effect via Rosenblatt process and Poisson jumps

&lt;abstract&gt;&lt;p&gt;This manuscript explores a new class of Hilfer fractional stochastic differential system, as driven by the Wiener process and Rosenblatt process through the application of non-instantaneous impulsive effects and Poisson jumps. Existence of a mild solution to the considered system is proved. Sufficient conditions for the controllability of the proposed control system are established. To prove our main results, we utilize fractional calculus, stochastic analysis, semigroup theory, and the Sadovskii fixed point theorem. In addition, to illustrate the theoretical findings, we present an example.&lt;/p&gt;&lt;/abstract&gt;

Approximate controllability of Atangana- Baleanu fractional stochastic differential systems with non-Gaussian process and impulses

Approximate controllability of Atangana- Baleanu fractional stochastic differential systems with non-Gaussian process and impulses

Stability and controllability of $$\psi $$-Caputo fractional stochastic differential systems driven by Rosenblatt process with impulses

Stability and controllability of $$\psi $$-Caputo fractional stochastic differential systems driven by Rosenblatt process with impulses

Optimal controls of impulsive fractional stochastic differential systems driven by Rosenblatt process with state‐dependent delay

AbstractIn this paper, we study a new class of noninstantaneous impulsive fractional stochastic differential systems driven by the Rosenblatt process with state‐dependent delay. We utilized the ‐resolvent family, fixed point technique, and solution operator to present the solvability of the proposed system. Further, we derived the existence of optimal multicontrol pairs for the considered system. Finally, the main results are validated with the aid of an example.

Approximate Controllability of Delayed Fractional Stochastic Differential Systems with Mixed Noise and Impulsive Effects

We herein report a new class of impulsive fractional stochastic differential systems driven by mixed fractional Brownian motions with infinite delay and Hurst parameter H^∈(1/2,1). Using fixed point techniques, a q-resolvent family, and fractional calculus, we discuss the existence of a piecewise continuous mild solution for the proposed system. Moreover, under appropriate conditions, we investigate the approximate controllability of the considered system. Finally, the main results are demonstrated with an illustrative example.

Approximate controllability of neutral fractional stochastic differential systems with control on the boundary

In this paper, the approximate boundary controllability results for neutral fractional stochastic differential system with fractional Brownian motion in Hilbert space is discussed. The existence mild solution of neutral fractional stochastic differential system with fractional Brownian motion is proved by using fractional calculus, compact semigroup, Schauder fixed-point theorem and stochastic analysis. The sufficient conditions for approximate boundary controllability of this system is established under the corresponding linear system is approximately controllable. Finally, an example is given to show the application of our results.

New Outcomes Regarding the Existence of Hilfer Fractional Stochastic Differential Systems via Almost Sectorial Operators

In this paper, we focus on the existence of Hilfer fractional stochastic differential systems via almost sectorial operators. The main results are obtained by using the concepts and ideas from fractional calculus, multivalued maps, semigroup theory, sectorial operators, and the fixed-point technique. We start by confirming the existence of the mild solution by using Dhage’s fixed-point theorem. Finally, an example is provided to demonstrate the considered Hilferr fractional stochastic differential systems theory.

Stability analysis of Atangana–Baleanu fractional stochastic differential systems with impulses

This paper is devoted to exploring a new class of Atangana–Baleanu fractional stochastic differential systems driven by fractional Brownian motion with non-instantaneous impulsive effects. Using resolvent family, fixed point technique, and fractional calculus, we analysed the existence and uniqueness of the mild solution. Moreover, we discussed the stability criteria for the proposed problem. A numerical example is given to illustrate the theoretical results.

Mean-square exponential stability of impulsive conformable fractional stochastic differential system with application on epidemic model

The main objective of this paper is to analyze the exponential stability of a new problem of an n-dimensional non-linear impulsive conformable stochastic differential system with the various fractional orders. By applying the conversion of the state variable, the solution of the aimed impulsive stochastic differential system is related to the corresponding solution of the non-impulsive system. Then the sufficient conditions for the exponential stability of the proposed impulsive system are accomplished by obtaining sufficient conditions for the exponential stability of the respective non-impulsive system. The novelty of this work is that the Razumikhin stability technique (without using the Lyapunov function) is used to achieve the stability conditions for the impulsive conformable stochastic system. The gained theoretical conclusions are confirmed by applying the conformable stochastic SIR epidemic model with impulsive immigration. Finally, the influence of fractional orders in the dynamical performance of the impulsive conformable stochastic SIR epidemic model is illustrated with graphical representations.

Existence and uniqueness of solutions of nonlinear fractional stochastic differential systems with nonlocal functional boundary conditions

The authors study the existence and uniqueness of solutions to nonlinear first-order fractional stochastic differential systems driven by Brownian motion and with nonlocal functional boundary conditions. The technique of proof involves Perov’s fixed point theorem with matrices that converge to zero and the Leray–Schauder theorem.

Approximate controllability of Sobolev-type nonlocal Hilfer fractional stochastic differential system

Approximate controllability of Sobolev-type nonlocal Hilfer fractional stochastic differential system

Approximate controllability of Sobolev-type nonlocal Hilfer fractional stochastic differential system

Approximate controllability of Sobolev-type nonlocal Hilfer fractional stochastic differential system

Some existence results for a stochastic differential system with non-Lipschitz conditions

The existence and uniqueness of mild solutions for a fractional stochastic differential system derived by Atangana–Baleanu fractional derivative have been discussed in a Hilbert space. A new set of results is obtained with non-Lipschitz conditions of nonlinear functions using the well-known Picard-type iterations, basic concepts of fractional calculus, sectorial operator, and stochastic analysis techniques. At last, an application is discussed to show the applicability of the developed theory.

Null controllability of nonlocal Sobolev-Type Hilfer fractional stochastic differential system driven by fractional Brownian motion and Poisson jumps

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.

Optimal controls for fractional stochastic differential systems driven by Rosenblatt process with impulses

AbstractThe objective of this article is to consider a new class of fractional stochastic differential systems driven by the Rosenblatt process with impulses. We used fractional calculus, stochastic analysis, and Krasnoselskii's fixed point theorem to study the existence of piecewise continuous mild solutions for the proposed system. Further, we discussed the existence of optimal controls for the considered system. Our main results are well supported by an illustrative example.

Optimal control for fractional stochastic differential system driven by fractional Brownian motion with Poisson jumps

Optimal control for fractional stochastic differential system driven by fractional Brownian motion with Poisson jumps