Controllability of impulsive neutral stochastic integro-differential systems driven by FBM with unbounded delay

In this paper we study the controllability results of impulsive neutral stochastic functional integrodifferential equations with infinite delay driven by fractional Brownian motion in a real separable Hilbert space. The controllability results are obtained by using stochastic analysis and a fixed-point strategy. In the end, one example is given to illustrate the feasibility and effectiveness of results obtained

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 $\hat{\cal H} \in ( 1/2, 1)$. 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.Comment: Please cite this paper as follows: Hakkar, N.; Dhayal, R.; Debbouche, A.; Torres, D.F.M. Approximate Controllability of Delayed Fractional Stochastic Differential Systems with Mixed Noise and Impulsive Effects. Fractal Fract. 2023, 7, 104. https://doi.org/10.3390/fractalfract702010

Stochastic differential equations with non-instantaneous impulses driven by a fractional Brownian motion

This paper is concerned with the existence and continuous dependence of mild solutions to stochastic differential equations with non-instantaneous impulses driven by fractional Brownian motions. Our approach is based on a Banach fixed point theorem and Krasnoselski-Schaefer type fixed point theorem.Ministerio de Economía y CompetitividadFondo Europeo de Desarrollo RegionalConsejería de Innovación, Ciencia y Empresa (Junta de Andalucía

Asymptotic behaviour of mild solution of nonlinear stochastic partial functional equations

This paper presents conditions to assure existence, uniqueness and stability for impulsive neutral stochastic integrodifferential equations with delay driven by Rosenblatt process and Poisson jumps. The Banach fixed point theorem and the theory of resolvent operator developed by Grimmer [R.C. Grimmer, Resolvent operators for integral equations in a Banach space, Trans. Am. Math. Soc., 273(1):333–349, 1982] are used. An example illustrates the potential benefits of these results

Impulsive neutral functional differential equations driven by a fractional Brownian motion with unbounded delay

In this paper, we prove the local and global existence and attractivity of mild solutions for stochastic impulsive neutral functional differential equations with infinite delay, driven by fractional Brownian motion.Fondo Europeo de Desarrollo RegionalMinisterio de Economía y CompetitividadJunta de Andalucí

On Asymptotic Stability of Stochastic Differential Equations with Delay in Infinite Dimensional Spaces

In most stochastic dynamical systems which describe process in engineering, physics and economics, stochastic components and random noise are often involved. Stochastic effects of these models are often used to capture the uncertainty about the operating systems. Motivated by the development of analysis and theory of stochastic processes, as well as the studies of natural sciences, the theory of stochastic differential equations in infinite dimensional spaces evolves gradually into a branch of modern analysis. In the analysis of such systems, we want to investigate their stabilities. This thesis is mainly concerned about the studies of the stability property of stochastic differential equations in infinite dimensional spaces, mainly in Hilbert spaces. Chapter 1 is an overview of the studies. In Chapter 2, we recall basic notations, definitions and preliminaries, especially those on stochastic integration and stochastic differential equations in infinite dimensional spaces. In this way, such notions as Q-Wiener processes, stochastic integrals, mild solutions will be reviewed. We also introduce the concepts of several types of stability. In Chapter 3, we are mainly concerned about the moment exponential stability of neutral impulsive stochastic delay partial differential equations with Poisson jumps. By employing the fixed point theorem, the p-th moment exponential stability of mild solutions to system is obtained. In Chapter 4, we firstly attempt to recall an impulsive-integral inequality by considering impulsive effects in stochastic systems. Then we define an attracting set and study the exponential stability of mild solutions to impulsive neutral stochastic delay partial differential equations with Poisson jumps by employing impulsive-integral inequality. Chapter 5 investigates p-th moment exponential stability and almost sure asymptotic stability of mild solutions to stochastic delay integro-differential equations. Finally in Chapter 6, we study the exponential stability of neutral impulsive stochastic delay partial differential equations driven by a fractional Brownian motion