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

Caraballo, Tomás; Ezzine, Faten; Hammami, Mohamed Ali

doi:10.1007/s10665-021-10164-w

Cited by 15 publications

(10 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We will study the asymptotic behaviors of the solutions of the stochastic perturbed system (2.2) in the sense that all state trajectories are bounded and approach a sufficiently small neighborhood of the origin. In this objective, we recall the following definitions, see [10,11,12,13].…”

Section: Linear Time-invariant Stochastic Perturbed Systemsmentioning

confidence: 99%

“…Remark 2.1. Different authors tackle the problem of practical stability of stochastic differential equations via Lyapunov functions, see [10,11,12,13,14]. The construction of appropriate Lyapunov functions is not always possible, which motivates us to look for another method.…”

Section: Linear Time-invariant Stochastic Perturbed Systemsmentioning

confidence: 99%

“…The Lyapunov stability has attracted the attention of many authors. There exists a huge amount of work on this subject, see [10,11,12,13,14]. However, the construction of a suitable Lyapunov function is still a difficult task.…”

Section: Introductionmentioning

confidence: 99%

“…The usual property of the solutions that can be deduced for such systems is ultimate boundedness. That means that the solutions remain in some neighborhoods of the origin after a sufficiently large time, (see [3], [10]- [13]). In different cases, the linearized system is independent of the control.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A new result on stabilization analysis for stochastic nonlinear affine systems via Gamidov’s inequality

Caraballo

Ezzine²,

Hammami³

2022

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

Self Cite

View full text Add to dashboard Cite

The Lyapunov approach is one of the most effective and efficient methods for the investigation of the stability of stochastic systems. Several authors analyzed the stability and stabilization of stochastic differential equations via Lyapunov techniques. Nevertheless, few results are concerned with the stability of stochastic systems based on the knowledge of the solution of the system explicitly. The originality of our work is to investigate the problem of stabilization of stochastic perturbed control-bilinear systems based on the explicit solution of the system by using the integral inequalities of the Gronwall type in particular Gamidov's inequality. Namely, under some restrictions on the perturbed term, and based on the method of integral inequalities, we prove that the stochastic system can be stabilized by constant feedback. Further, we study the problem of stabilization of stochastic perturbed control affine systems based on the use of bilinear approximation. Different examples are provided to verify the effectiveness of the proposed results.

show abstract

Section: Linear Time-invariant Stochastic Perturbed Systemsmentioning

confidence: 99%

Section: Linear Time-invariant Stochastic Perturbed Systemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A new result on stabilization analysis for stochastic nonlinear affine systems via Gamidov’s inequality

Caraballo

Ezzine²,

Hammami³

2022

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

Self Cite

View full text Add to dashboard Cite

show abstract

“…This property is referred to as practical stability. Several results on the stability of the nontrivial solution re proposed in [1,2,10,11,12]. To our knowledge, no work has been reported about the practical stability of linear time-varying singular systems.…”

Section: Introductionmentioning

confidence: 99%

Growth conditions for the stability of a class of time-varying perturbed singular systems

Ezzine¹,

Hammami²

2022

Kybernetika

View full text Add to dashboard Cite

In this paper, we investigate the problem of stability of linear time-varying singular systems, which are transferable into a standard canonical form. Sufficient conditions on exponential stability and practical exponential stability of solutions of linear perturbed singular systems are obtained based on generalized Gronwall inequalities and Lyapunov techniques. Moreover, we study the problem of stability and stabilization for some classes of singular systems. Finally, we present a numerical example to validate the effectiveness of the abstract results of this paper.

show abstract

Estimates of Exponential Convergence for Solutions of Stochastic Nonlinear Systems

2023

Self Cite

View full text Add to dashboard Cite

This paper aims to analyze the behavior of the solutions of a stochastic perturbed system with respect to the solutions of the stochastic unperturbed system. To prove our stability results, we have derived a new Gronwall-type inequality instead of the Lyapunov techniques, which makes it easy to apply in practice and it can be considered as a more general tool in some situations. On the one hand, we present sufficient conditions ensuring the global practical uniform exponential stability of SDEs based on Gronwall's inequalities. On the other hand, we investigate the global practical uniform exponential stability with respect to a part of the variables of the stochastic perturbed system by using generalized Gronwall's inequalities. It turns out that, the proposed approach gives a better result comparing with the use of a Lyapunov function. A numerical example is presented to illustrate the applicability of our results.

show abstract

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

Cited by 15 publications

References 17 publications

A new result on stabilization analysis for stochastic nonlinear affine systems via Gamidov’s inequality

A new result on stabilization analysis for stochastic nonlinear affine systems via Gamidov’s inequality

Growth conditions for the stability of a class of time-varying perturbed singular systems

Estimates of Exponential Convergence for Solutions of Stochastic Nonlinear Systems

Contact Info

Product

Resources

About