Impulsive Stochastic Differential Equations Research Articles

Solvability and optimal controls of non-instantaneous impulsive stochastic fractional differential equation of order q ∈ (1,2)

In this article, we study the stochastic fractional optimal control problem for a system governed by a class of non-instantaneous impulsive stochastic fractional differential equation in the infinite-dimensional spaces. We utilize the fractional calculus, stochastic analysis theory and a suitable fixed point approach to present the solvability of the stochastic fractional system. Then, the existence of optimal state-control pairs of the Lagrange problem is derived without uniqueness of solutions of the stochastic system. Finally, the main results are validated with the aid of an example.

On stability of stochastic differential equations with random impulses driven by Poisson jumps

In this article, we study the existence of solutions and their stability of random impulsive stochastic functional differential equations (ISFDEs) driven by Poisson jumps with finite delays. Initially, we prove the existence of the mild solutions to the equations by using Banach fixed point theorem. Then, we study the stability of random ISFDEs through the continuous dependence of solutions on initial condition. Next, we investigate the Hyers Ulam stability results under the Lipschitz condition on a bounded and closed interval. Finally, an example is presented to illustrate our results.

Averaging principle for impulsive stochastic partial differential equations

This paper focuses on systems of stochastic partial differential equations with impulse effects. We establish an averaging principle such that the solution to the complex original nonlinear impulsive stochastic evolution equations can be approximated by that to the more simplified averaged stochastic evolution equations without impulses. By adopting stochastic analysis theory, semigroup approach and inequality technique, sufficient conditions are formulated and the mean square convergence is proved. This ensures that we can concentrate on the averaged system instead of the original system, thus providing a solution for reduction of complexity.

Existence, Uniqueness and Stability of Impulsive Stochastic Partial Neutral Functional Differential Equations with Infinite Delays Driven by a Fractional Brownian Motion

This article presents the result on existence, uniqueness and stability of mild solution of impulsive stochastic partial neutral functional differential equations driven by a fractional Brownian motion. The results are obtained by using the method of successive approximation and Bihari’s inequality.

Existence and Stability Results of Impulsive Stochastic Partial Neutral Functional Differential Equations with Infinite Delays and Poisson Jumps

In this paper, we are focused upon the results on existence, uniqueness and stability of mild solution of impulsive stochastic partial neutral functional differential equations with infinite delays and poisson jumps. The results are obtained by using the method of successive approximation and Bihari’s inequality.

Stepanov-like pseudo almost periodicity in distribution and optimal control to impulsive partial stochastic differential equations

ABSTRACT This paper concerns the Stepanov-like pseudo almost periodicity in distribution for a stochastic process and optimal control. Using the analytic semigroup theory, fractional power operators, stochastic analysis theory and fixed point strategy, we obtain the existence of Stepanov-like pseudo almost periodic in distribution mild solutions for impulsive partial stochastic functional differential equations under non-Lipschitz conditions. The existence of optimal pairs of system governed by impulsive partial stochastic control systems is also presented. Finally, an example is given for demonstration.

Impulsive stochastic fractional differential equations driven by fractional Brownian motion

In this research, we study the existence and uniqueness results for a new class of stochastic fractional differential equations with impulses driven by a standard Brownian motion and an independent fractional Brownian motion with Hurst index 1/2< H<1 under a non-Lipschitz condition with the Lipschitz one as a particular case. Our analysis depends on an approximation scheme of Carathéodory type. Some previous results are improved and extended.

Stability of a Class of Impulsive Neutral Stochastic Functional Partial Differential Equations

In this paper, a class of impulsive neutral stochastic functional partial differential equations driven by Brownian motion and fractional Brownian motion is investigated. Under some suitable assumptions, the pth moment exponential stability is discussed by means of the fixed-point theorem. Our results also improve and generalize some previous studies. Moreover, one example is given to illustrate our main results.

Periodic averaging method for impulsive stochastic dynamical systems driven by fractional Brownian motion under non-Lipschitz condition

This paper presents the periodic averaging principle for impulsive stochastic dynamical systems driven by fractional Brownian motion (fBm). Under non-Lipschitz condition, we prove that the solutions to impulsive stochastic differential equations (ISDEs) with fBm can be approximated by the solutions to averaged SDEs without impulses both in the sense of mean square and probability. Finally, an example is provided to illustrate the theoretical results.

Existence, uniqueness and stability of impulsive stochastic neutral functional differential equations driven by Rosenblatt process with varying-time delays

Abstract Hermite processes are self-similar processes with stationary increments; the Hermite process of order 1 is fractional Brownian motion (fBm) and the Hermite process of order 2 is the Rosenblatt process. In this paper we consider a class of impulsive neutral stochastic functional differential equations with variable delays driven by Rosenblatt process with index {H\in(\frac{1}{2},1)} , which is a special case of a self-similar process with long-range dependence. More precisely, we prove an existence and uniqueness result, and we establish some conditions, ensuring the exponential decay to zero in mean square for the mild solution by means of the Banach fixed point theory. Finally, an illustrative example is given to demonstrate the effectiveness of the obtained result.

Existence and mean-square exponential stability of mild solutions for impulsive stochastic partial differential equations with noncompact semigroup

In this paper, with the help of the Hausdorff measure of noncompactness, the Mönch fixed point theorem and some inequality technique, some new criteria to guarantee the mild solution for impulsive stochastic partial differential equations with varying-time delays are obtained. Then, by establishing a new impulsive-integral inequality, some sufficient conditions to ensure mean-square exponential stability of mild solution for impulsive stochastic partial differential equations with varying-time delays are obtained which will significantly improve some previous results [5,20]. At last, we give an example to illustrate the effectiveness and feasibility of the obtain results in our paper.

Stability analysis of impulsive stochastic functional differential equations

The paper discusses both p-th moment and almost sure exponential stability of solutions to stochastic functional differential equations with impulsive by using the Razumikhin-type technique. The main goal is to find some conditions that could be applied to control more easily than using the usual method with Lyapunov functionals. Examples are also given to illustrate the efficiency of the obtained results.

Existence and Stability Results for Impulsive Stochastic Functional Integrodifferential Equation with Poisson Jumps

In this article we present the existence, uniqueness and stability of mild solutions for impulsive stochastic functional integro differential equations with non-Lipschitz condition. The mild solution is obtained by using a resolvent operator in a different sense and the results are proved by using the method of successive approximation and Bihari’s inequality.

Stochastic stabilization using aperiodically sampled measurements

This paper addresses the stabilization problem of sector-bounded nonlinear systems with sampled measurements via discrete-time stochastic feedback control. Unlike the previous studies, the closed-loop system is modeled as an impulsive stochastic differential equation. By developing a quasi-periodic polynomial Lyapunov function and sampling-time-dependent Lyapunov function based methods, two sufficient conditions for almost sure exponential stability are derived in terms of differential matrix inequalities (DMIs) and linear matrix inequalities (LMIs). It is shown that the DMI-based conditions can be formulated as a sum of squares (SOSs). Moreover, the obtained results are adapted to sampled-data stochastic/deterministic systems. The numerical examples illustrate the theoretical results.

On the pth moment integral input‐to‐state stability and input‐to‐state stability criteria for impulsive stochastic functional differential equations

SummaryIn this paper, several new Razumikhin‐type theorems for impulsive stochastic functional differential equations are studied by applying stochastic analysis techniques and Razumikhin stability approach. By developing a new comparison principle for stochastic version, some novel criteria of the pth moment integral input‐to‐state stability and input‐to‐state stability are derived for the related systems. The feature of the criteria shows that time‐derivatives of the Razumikhin functions are allowed to be indefinite, even unbounded, which can loosen the constraints of the existing results. Finally, some examples are given to illustrate the usefulness and significance of the theoretical results.

Existence results systems coupled impulsive neutral stochastic functional differential equations with the measure of noncompactness

This paper is devoted to study the existence of solutions for a class of mild solutions for a class of impulsive neutral stochastic functional differential equations driven fractional Brownian motion (fBm) with noncompact semigroup in Hilbert spaces. The new results are obtained by using the Hausdorff measure of noncompactness. The arguments are based upon Monch’s fixed point theorem. Finally, an example is provided to illustrate the developed theory.

Optimal controls for impulsive partial stochastic differential equations with weighted pseudo almost periodic coefficients

In this paper, we study optimal controls of a class of impulsive partial stochastic differential equations with weighted pseudo almost periodic coefficients in Hilbert spaces. Firstly, using interpolation theory, the stochastic analysis techniques and a fixed-point theorem, the existence of p-mean piecewise weighted pseudo almost periodic in distribution mild solutions for the impulsive stochastic control system is presented. Secondly, the existence of optimal pairs of system governed by impulsive partial stochastic differential equations is also obtained. Finally, application to optimal control problems of impulsive parabolic control system is considered.

Approximate Controllability for a Class of Non-instantaneous Impulsive Stochastic Fractional Differential Equation Driven by Fractional Brownian Motion

In this manuscript, we study the approximate controllability for a class of non-instantaneous impulsive stochastic fractional differential equation driven by fractional Brownian motion in Hilbert space. The results are obtained by using the $$\rho $$ -resolvent family, and fixed point techniques. Finally, we have given an example to illustrate the applicability of our results.

Stability of square-mean almost automorphic mild solutions to impulsive stochastic differential equations driven by G-Brownian motion

ABSTRACT In this paper, we consider a class of impulsive stochastic differential equations driven by G-Brownian motion (IGSDEs, in short). In particular, we study the exponentially and quasi sure exponential stability of the square-mean almost automorphic solutions of the IGSDEs. Specifically by employing the Lyapunov function method, a new set of sufficient conditions is derived for obtaining the required result.

Periodic averaging method for impulsive stochastic differential equations with Lévy noise

The purpose of this paper is to present a periodic averaging method for impulsive stochastic differential equations with Lévy noise under non-Lipschitz condition. It is shown that the solutions of impulsive stochastic differential equations with Lévy noise converge to the solutions of the corresponding averaged stochastic differential equations without impulses