Exponential Stability Of Mild Solutions Research Articles

Exponential stability results for second-order impulsive neutral stochastic differential equations

This paper is aimed to discuss the existence and exponential stability of mild solutions for second-order impulsive neutral stochastic differential equations (INSDEs). Firstly, we derive the existence of mild solutions for INSDEs by using the Banach fixed point theorem. Then, the exponential stability in the pth moment of mild solution for the considered INSDEs is proved by means of the generalized impulsive-integral inequality, which can effectively improve some known existing ones. Finally, as an application, an example is given to illustrate the efficiency of the obtained theoretical results.

Trajectory control and [formula omitted]th moment exponential stability of neutral functional stochastic systems driven by Rosenblatt process

The purpose of this paper is to determine a class of neutral stochastic functional integro-differential system in real separable Hilbert spaces as well as the exponential stability results. The Rosenblatt process acts as the driving force behind the systems. Initially, the existence results for mild solutions of the stochastic system is investigated by using stochastic analysis, integro-differential theory, and fixed point theory. The analysis of the exponential stability of mild solutions to nonlinear neutral stochastic integro-differential systems driven by Rosenblatt process is the main objective of the investigation’s further stages. The system’s trajectory controllability is then examined using Gronwall’s inequality. An example is given to validate the results at the end. Our work extends the work of Chalishajar et al. (2010), Chalishajar and Chalishajar (2015), Chalishajar et al. (2023), Muslim and George (2019) and Durga et al. (2022) where the pth moment exponential stability has not discussed. Also, numerical simulation has not studied in Chalishajar et al. (2023), Muslim and George (2019) and Durga et al. (2022).

Some results on the existence and stability of impulsive delayed stochastic differential equations with Poisson jumps

<abstract><p>This paper is concerned with the existence, uniqueness and exponential stability of mild solutions for a class of impulsive stochastic differential equations driven by Poisson jumps and time-varying delays. Utilizing the successive approximation method, we obtain the criteria of existence and uniqueness of mild solutions for the considered impulsive stochastic differential equations. Then, the exponential stability in the $ p $th moment of the mild solution is also devised for considered equations by establishing an improved impulsive-integral inequality, which improves some known existing ones. Finally, an example and numerical simulations are given to illustrate the efficiency of the obtained theoretical results.</p></abstract>

$ p $th moment exponential stability and convergence analysis of semilinear stochastic evolution equations driven by Riemann-Liouville fractional Brownian motion

<abstract><p>Many works have been done on Brownian motion or fractional Brownian motion, but few of them have considered the simpler type, Riemann-Liouville fractional Brownian motion. In this paper, we investigate the semilinear stochastic evolution equations driven by Riemann-Liouville fractional Brownian motion with Hurst parameter $ H < 1/2 $. First, we prove the $ p $th moment exponential stability of mild solution. Then, based on the maximal inequality from Lemma 10 in <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>, the uniform boundedness of $ p $th moment of both exact and numerical solutions are studied, and the strong convergence of the exponential Euler method is established as well as the convergence rate. Finally, two multi-dimensional examples are carried out to demonstrate the consistency with theoretical results.</p></abstract>

Existence and Exponential Stability for Neutral Impulsive Stochastic Integrodifferential Equations with Fractional Brownian Motion Driven by Poisson Jumps

In this paper, we establish the results on existence and uniqueness of mild solution of neutral impulsive stochastic integrodifferential equations with fractional Brownian motion driven by Poisson jumps in a Hilbert space. Further, by using an impulsive integral inequality, we study the new sufficient conditions that ensure the exponential stability of mild solution in the mean square moment by utilizing the fractional power of operators and the semigroup theory.

Exponential behavior of neutral impulsive stochastic integro-differential equations driven by Poisson jumps and Rosenblatt process

AbstractIn this article, we are concerned with the neutral impulsive stochastic integro-differential equations driven by Poisson jumps and Rosenblatt process. By using resolvent operator and some analysis techniques, we ensure existence and uniqueness of solutions. Further, we investigate exponential stability of mild solutions. We have also given an example to illustrate our theoretical results.

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.

Controllability and stability of fractional stochastic functional systems driven by Rosenblatt process

In this paper, we investigate the controllability and exponential stability results for a class of nonlinear neutral stochastic functional differential control systems in the presence of infinite delay driven by Rosenblatt process in real separable Hilbert spaces. Firstly, by using stochastic analysis approach, fractional calculus theory and a fixed point technique, we prove the controllability result for the mild solutions of the nonlinear control systems with the condition that the associated linear system is controllable. Secondly, we discuss the exponential stability of mild solutions of the nonlinear neutral stochastic differential systems driven by Rosenblatt process. Finally, an example is presented to illustrate the obtained theory.

The existence and exponential stability of random impulsive fractional differential equations

In this paper, the existence and exponential stability of mild solutions of random impulsive fractional differential equations are studied. The results are obtained by using the Leray–Schauder alternative fixed point theorem.

Existence and exponential stability for impulsive neutral stochastic functional differential equations driven by fBm with noncompact semigroup via Mönch fixed point

In this paper, we investigate the existence and exponential stability of mild solutions for a class of impulsive neutral stochastic functional differential equations driven by fBm with noncompact semigroup in Hilbert spaces. Sufficient conditions for the existence of mild solutions are obtained using the Hausdorff measure of noncompactness and the Mönch fixed point theorem. Further, we establish a new impulsive-integral inequality to prove the exponential stability of mild solutions in the mean square moment. Finally, an example is presented to illustrate our obtained results.

Existence and exponential stability of a class of impulsive neutral stochastic partial differential equations with delays and Poisson jumps

In this paper, the existence, uniqueness and exponential stability of mild solution for a class of impulsive neutral stochastic partial differential equations with delays and Poisson jumps are studied. The existence and uniqueness of mild solution are studied by means of successive approximations, and the exponential stability in pth moment of mild solution is investigated by employing an appropriate impulsive-integral inequality. An example is given to illustrate our main results.

Exponential stability behavior of neutral stochastic integrodifferential equations with fractional Brownian motion and impulsive effects

In this paper, by employing the fractional power of operators, semigroup theory, and fixed point strategy we obtain some new criteria ensuring the existence and exponential stability of a class of impulsive neutral stochastic integrodifferential equations driven by a fractional Brownian motion. We establish some new sufficient conditions that ensure the exponential stability of mild solution in the mean square moment by utilizing an impulsive integral inequality. Also, we provide an example to show the efficiency of the obtained theoretical result.

Existence and exponential stability for impulsive stochastic partial functional differential equations

In this paper, the existence and uniqueness, the exponential stability, and the almost sure exponential stability of mild solution for impulsive stochastic partial functional differential equations with finite delay are considered. Some sufficient conditions are established for our concerned problems, and some existing results are generalized and improved. Finally, an illustrative example is provided to show the feasibility and effectiveness of the obtained results.

On exponential stability of mild solutions for some stochastic partial integrodifferential equations

In this work, we study the existence and exponentially stability in p-mean square for some stochastic integrodifferential equation with delays. Also pathwise exponential stability is proved for p>2. We assume that the linear part has a resolvent operator in the sense given by Grimmer (1982). The delayed part is assumed to be continuous. Our results are proved by using the stochastic convolution.

Existence and stability results for semilinear systems of impulsive stochastic differential equations with fractional Brownian motion

ABSTRACTSome results on the existence and uniqueness of mild solution for a system of semilinear impulsive differential equations with infinite fractional Brownian motions are proved. The approach is based on Perov's fixed point theorem and a new version of Schaefer's fixed point theorem in generalized Banach spaces. The relationship between mild and weak solutions and the exponential stability of mild solutions are investigated as well. The abstract theory is illustrated with an example.

Existence and exponential stability for neutral stochastic integrodifferential equations with impulses driven by a fractional Brownian motion

In this paper, we establish the results on existence and uniqueness of mild solution of impulsive neutral stochastic integrodifferential equations driven by a fractional Brownian motion. Further, by using an impulsive integral inequality, some novel sufficient conditions are derived to ensure the exponential stability of mild solution in the mean square moment. The results are obtained by utilizing the fractional power of operators and the semigroup theory. Finally, an example is presented to demonstrate the effectiveness of the proposed result.

On Yosida Approximations of McKean–Vlasov Type Stochastic Evolution Equations

This article is concerned with a semilinear McKean–Vlasov type stochastic evolution equation in a real Hilbert space. The main goal of the article is to study the existence and uniqueness of mild solutions, Yosida approximations of mild solutions of such equations, and to deduce the weak convergence of the corresponding induced probability measures. As an application, we also study the exponential stability of mild solutions of such equations.

Exponential stability for second-order neutral stochastic differential equations with impulses

In this paper, we investigate the problem on the exponential stability of mild solution for the second-order neutral stochastic partial differential equations with impulses by utilising the cosine function theory. A set of novel sufficient conditions is derived by establishing an impulsive integral inequality. As a final point, an example is given to illustrate the effectiveness of the obtained theory.

Local stability and parameter dependence of mild solutions for stochastic differential equations

For nonlinear stochastic equations dx(t )=( Ax(t )+ f (t, x(t), λ)) dt + g(t, x(t), λ) dω(t )w ith parameter λ in a Hilbert space, we show the existence and uniqueness of mild solutions. Provided that f satisfies a locally Lipschitz condition and g is a uniformly Lipschitz function, some sufficient conditions for p (p ≥ 2) moment locally exponential stability as well as almost surely exponential stability of mild solutions are obtained under a sufficiently small initial value ξ . Meanwhile, we also consider parameter dependence of stable mild solutions for the stochastic system if f , g are sufficiently small Lipschitz perturbations in the parameter λ.

Robust exponential stability of mild solutions to impulsive stochastic neutral integro-differential equations with delay

Robust exponential stability of mild solutions to impulsive stochastic neutral integro-differential equations with delay