Stability Of Stochastic Differential Equations Research Articles

Partial stability analysis of linear time-varying perturbed stochastic systems via a refined integral inequality

The Lyapunov approach is one of the most effective and efficient means of studying the partial stability of stochastic systems. A number of authors have analysed the partial practical stability of stochastic differential equations using Lyapunov techniques. Nevertheless, no results are concerned with the partial stability of stochastic systems based on the knowledge of the solution of the system explicitly. This paper has been concerned with the problem of partial practical stability for linear time-varying stochastic perturbed systems. Necessary and sufficient conditions for partial practical uniform exponential stability are given based on generalised Gronwall inequalities, in particular of Gamidov's type. An example is developed to illustrate the obtained results.

Almost sure polynomial stability and stabilization of stochastic differential systems with impulsive effects

In this paper, we first consider the almost surely and pth moment polynomial stability of stochastic differential equations with impulsive effects. Sufficient conditions are given to obtain the polynomial stability. Then the stabilization of stochastic differential systems is studied, which shows that impulse can make stochastic differential equations stable.

Fuzzy Mittag–Leffler–Hyers–Ulam–Rassias Stability of Stochastic Differential Equations

Stability is the most relevant property of dynamical systems. The stability of stochastic differential equations is a challenging and still open problem. In this article, using a fuzzy Mittag–Leffler function, we introduce a new fuzzy controller function to stabilize the stochastic differential equation (SDE) ν′(γ,μ)=Fγ,μ,ν(γ,μ). By adopting the fixed point technique, we are able to prove the fuzzy Mittag–Leffler–Hyers–Ulam–Rassias stability of the SDE.

Stochastic calculus for tempered fractional Brownian motion and stability for SDEs driven by TFBM

The objective of this article is to introduce and study Itô type stochastic integrals with respect to tempered fractional Brownian motion (TFBM) of Hurst index and tempering parameter , by using the Wick product. The main tools are fractional calculus and Malliavin calculus. The Itô formula for this stochastic integral is established for the Itô type processes driven by TFBM. Based on this new Itô formula, we analyze the stability of stochastic differential equations driven by TFBM in the sense of -th moment. A numerical example is given to illustrate our stability results.

On the Practical Exponential Stability in Mean Square of Stochastic Perturbed Systems Via a Lyapunov Approach

The Lyapunov method is one of the most effective methods to analyze the stability of stochastic differential equations (SDEs). Different authors analyzed the stability of SDEs based on Lyapunov techniques when the origin can be considered as an equilibrium point. When the origin is not necessarily an equilibrium point, it is still possible to analyze the asymptotic stability of solutions concerning a small neighborhood of the origin. The purpose is to study the asymptotic stability of a system whose solution behavior is a small ball of state space or close to it. Thus, all state trajectories are bounded and close to a sufficiently small neighborhood of the origin. In this sense, the limited boundedness of solutions of random systems, or the chance of convergence of solutions needs to be analyzed on a ball centered on the origin. This is the so called “Practical Stability”. In this article, we mainly investigate the practical uniform exponential stability in the mean square of stochastic linear time–invariant systems. In addition, we are developing the problem of stabilization of certain classes of perturbed stochastic systems. Our crucial techniques include Lyapunov techniques and generalized Gronwall inequalities. Lastly, we provide a numerical example to illustrate our theoretical findings.

Rapid stabilization of stochastic quantum systems in a unified framework

Rapid stabilization of general stochastic quantum systems is investigated based on the rapid stability of stochastic differential equations. We introduce a Lyapunov–LaSalle-like theorem for a class of nonlinear stochastic systems first, based on which a unified framework of rapidly stabilizing stochastic quantum systems is proposed. According to the proposed unified framework, we design the switching state feedback controls to achieve the rapid stabilization of single-qubit systems, two-qubit systems, and N-qubit systems. From the unified framework, the state space is divided into two state subspaces, and the target state is located in one state subspace, while the other system equilibria are located in the other state subspace. Under the designed state feedback controls, the system state can only transit through the boundary between the two state subspaces no more than two times, and the target state is globally asymptotically stable in probability. In particular, the system state can converge exponentially in (all or part of) the state subspace where the target state is located. Moreover, the effectiveness and rapidity of the designed state feedback controls are shown in numerical simulations by stabilizing GHZ states for a three-qubit system.

PARTIAL PRACTICAL STABILITY AND ASYMPTOTIC STABILITY OF STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY LÉVY NOISE WITH A GENERAL DECAY RATE

In this paper, we mainly study the almost sure partial practical stability of stochastic differential equations driven by Lévy noise with a general decay rate. By establishing a suitable Lyapunov function and using Exponential Martingale inequality and Borel-Cantelli theorem, giving some sufficient conditions that can guarantee the almost sure partial practical stability of equations. At the same time, we also study general conditions that guarantee the almost sure asymptotic stability of the equation. Finally, we also give two examples to illustrate our theoretical results.

Asymptotic stability of stochastic differential equations driven by G-Lévy process with delay feedback control

Given an unstable stochastic differential equations, the stabilisation by delay feedback controls for such equations under Lipschitz conditions or highly nonlinear conditions have been discussed by several authors. However, there is few works on the stabilisation by delay feedback controls under the sub-linear expectation associated with a G-L?vy process. The aim of this paper is to design delay feedback controls in the drift part and obtain the asymptotical stability in mean square and quasi-surely asymptotical stability for the stochastic differential equations driven by G-L?vy process with the polynomial growth condition. Lastly, we give an example to verify the obtained theory.

The Improved Stability Analysis of Numerical Method for Stochastic Delay Differential Equations

In this paper, the improved split-step θ method, named the split-step composite θ method, is proposed to study the mean-square stability for stochastic differential equations with a fixed time delay. Under the global Lipschitz and linear growth conditions, it is proved that the split-step composite θ method with θ≥0.5 shows mean-square stability. An approach to improving numerical stability is illustrated by choices of parameters of this method. Some numerical examples are presented to show the accordance between the theoretical and numerical results.

Exponential ultimate boundedness and stability of stochastic differential equations with impulese

AbstractThe paper mainly studies globally pth moment exponentially ultimate boundedness and pth moment exponential stability of impulsive stochastic functional differential equations. By using the Lyapunov direct method of Razumikhin‐type condition and the principle of comparison, some sufficient conditions for globally pth moment exponentially ultimate boundedness and globally pth moment exponential stability are presented. Theorems require the linear coefficients of the upper bound of Lyapunov differential operators are time‐varying functions; this generalizes the previous results. When the time delay is not considered in the system, a unified criterion is given to achieve boundedness and stability when the system is disturbed by stabilizing impulse and destabilizing impulse. It shows that the stochastic differential equation may be unbounded or unstable, and it can be bounded or stable by adding appropriate impulsive perturbation. Finally, we use two examples to illustrate the validity of our results.

Stability with respect to a part of the variables of stochastic nonlinear systems driven by G-Brownian motion

In this paper, we investigate the pth moment exponential stability of stochastic differential equations driven by G-Brownian motion (G-SDEs) with respect to a part of the variables by means of the G-Lyapunov functions and recently developed Itô's calculus for SDEs driven by G-Brownian motion, as well as Gronwall's inequalities. We establish sufficient conditions to ensure the quasi sure exponential stability of stochastic differential equations perturbed by G-Brownian motion with respect to a part of the variables. Some illustrative examples to show the usefulness of the stability with respect to a part of the variables notion are also provided.

Exponential stability of stochastic systems: A pathwise approach

We provide a pathwise approach using semigroup technique to study the asymptotic stability for stochastic differential equations which admit a unique equilibrium. The driving noises in consideration are [Formula: see text] — Hölder continuous with [Formula: see text], so that the perturbed systems can be solved using rough path theory, where the rough integrals are interpreted in the Gubinelli sense for controlled rough paths. Our approach suggests an alternative method for stochastic systems with standard Brownian noises, by not using Itô formula but a relaxed isometry property of Itô stochastic integrals.

Stability of Hybrid SDEs Driven by fBm

In this paper, the exponential stability of stochastic differential equations driven by multiplicative fractional Brownian motion (fBm) with Markovian switching is investigated. The quasi-linear cases with the Hurst parameter H ∈ (1/2, 1) and linear cases with H ∈ (0, 1/2) and H ∈ (1/2, 1) are all studied in this work. An example is presented as a demonstration.

Asymptotic stability of fractional order (1,2] stochastic delay differential equations in Banach spaces

• In this article we have studied the asymptotic stability and mean square stability of fractional-order (1,2] stochastic differential equations (SDEs). • We have considered the family of SDEs with variable delay in state. • For obtaining main results we used Banach fixed point theorem and imposed Lipschitz condition on nonlinearity. • Results are advanced and weighted enough as a contribution to stability theory of fractional SDEs. In this article, we discuss the asymptotic stability and mean square stability of stochastic differential equations of fractional-order 1 < α ≤ 2 . We have considered the family of stochastic differential equations with variable delay in the state. For proving our main results, we apply the Banach fixed point theorem and imposed the Lipschitz condition on nonlinearity. Finally, we present an example to illustrate the obtained theory.

Partial stability analysis of stochastic differential equations with a general decay rate

This paper is concerned with the almost sure partial practical stability of stochastic differential equations with general decay rate. We establish some sufficient conditions based upon the construction of appropriate Lyapunov functions. Finally, we provide a numerical example to demonstrate the efficiency of the obtained results.

Stabilization of Stochastic Differential Equations Driven by G-Brownian Motion with Aperiodically Intermittent Control

The paper is devoted to studying the exponential stability of a mild solution of stochastic differential equations driven by G-Brownian motion with an aperiodically intermittent control. The aperiodically intermittent control is added into the drift coefficients, when intermittent intervals and coefficients satisfy suitable conditions; by use of the G-Lyapunov function, the p-th exponential stability is obtained. Finally, an example is given to illustrate the availability of the obtained results.

Delay feedback stabilisation of stochastic differential equations driven by G-Brownian motion

This paper aims to design the feedback control based on past states to stabilise a class of nonlinear stochastic differential equations driven by G-Brownian motion. By building up the connection between the delay feedback control and the control function of no-delay, sufficient conditions are established in terms of size of the time delay ensuring both the pth moment exponential stability and quasi-surely exponential stability of the delay feedback controlled system. Moreover, methods for determining the upper bound of the length of the time delay and further designing the delay feedback controller are provided. A numerical example is presented to demonstrate our new theory.

Stability of stochastic differential equations driven by multifractional Brownian motion

Abstract Our aim in this paper is to establish some strong stability results for solutions of stochastic differential equations driven by a Riemann–Liouville multifractional Brownian motion. The latter is defined as a Gaussian non-stationary process with a Hurst parameter as a function of time. The results are obtained assuming that the pathwise uniqueness property holds and using Skorokhod’s selection theorem.

Further results on stabilization of stochastic differential equations with delayed feedback control under $ G $-expectation framework

&lt;p style='text-indent:20px;'&gt;In this paper we establish a comparison approach to study stabilization of stochastic differential equations driven by &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ G $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-Brownian motion with delayed (&lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ G $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-SDDEs for short) feedback control. This theory also extends to a general range of moment order and brings more choices of &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ p $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. Finally, a simple example is proposed to demonstrate the applications of our theory.&lt;/p&gt;

Stability of stochastic differential equations driven by the time-changed Lévy process with impulsive effects

The stability of nonlinear stochastic differential equations driven by time-changed Lévy process with impulsive effects is discussed in this paper. Some sufficient conditions are provided to guarantee the solutions to be stable in different senses. The stochastic perturbation is also investigated for some unstable time-changed differential equations with impulses. The efficiency of the proposed results is illustrated by some examples with numerical simulations.