Non-linear integral inequalities and applications to asymptotic stability

Hammi, Mekki; Hammami, Mohamed

doi:10.1093/imamci/dnu016

Cited by 10 publications

(12 citation statements)

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“…Lemma 1 (Gronwall-Bellman's inequality 33 ). Let x and be nonnegative continuous functions on R + satisfying the inequality…”

Section: Integral Inequalitiesmentioning

confidence: 99%

“…30 Another approach to study the stability of perturbed systems consists of using the behavior of the solutions of the associated unperturbed system in the vicinity of the equilibrium point. [31][32][33][34][35][36][37] Being formulated in terms of integral inequalities of Gronwall type, it is a type of stability which is easy to verify in practice, and it extends the class of systems for which the effect of perturbations can be measured. In this article, we make use of such an approach in order to derive some conditions that ensures the (global) uniform exponential stability of perturbed systems.…”

Section: Introductionmentioning

confidence: 99%

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New results on the uniform exponential stability of nonautonomous perturbed dynamical systems

Benjemaa

Gouadri

Hammami

2021

Intl J Robust & Nonlinear

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In this article, we investigate the asymptotic behaviors of the solutions of nonlinear dynamic systems nearby an equilibrium point, when the nominal parts are subject to non-necessarily small perturbations. We show that, under some estimates on the perturbation term, the equilibrium point remains (globally) uniformly exponentially stable. The obtained stability results can easily be applied in practice since they are based on the Gronwall-Bellman inequalities rather than the classical Lyapunov methods that require the knowledge of a Lyapunov function. Several numerical examples, as well as an application to control and mechanical systems, are given in illustration.

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“…Lemma 1 (Gronwall-Bellman's inequality 33 ). Let x and be nonnegative continuous functions on R + satisfying the inequality…”

Section: Integral Inequalitiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

New results on the uniform exponential stability of nonautonomous perturbed dynamical systems

Benjemaa

Gouadri

Hammami

2021

Intl J Robust & Nonlinear

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“…Later on, the integral form was proved in 1943 by the American mathematician Bellmen [8]. Since then, many authors introduced several generalizations of this inequality, see [6,7,15,20,22].…”

Section: Introductionmentioning

confidence: 99%

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.

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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.

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“…In mathematics, this kind of inequalities allows one to bound a function that is known to satisfy a certain differential or integral inequality by the solution to the corresponding differential or integral equation. This type of inequalities can be used in the study of existence, uniqueness and stability properties of solutions 10 H. DAMAK, M. A. HAMMAMI and A. KICHA to differential perturbed equations (see [8,9]). This work focuses on the study of the global uniform h-stability and the global uniform boundedness of time-varying perturbed differential equations under conditions on the perturbed terms and we will see how we can use the Gronwall's inequalities in this research.…”

Section: Introductionmentioning

confidence: 99%

h-stability and boundedness results for solutions to certain nonlinear perturbed systems

Damak¹,

Hammani²,

Kicha³

2021

Math. Appl.

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In this work, we establish some new sufficient conditions to show the uniform h-stability for nonlinear time-varying perturbed systems which can be viewed as an extension of the uniform exponential stability and polynomial stability. Also, we study the boundedness of solutions when the origin is not necessarily an equilibrium point of the perturbed system. The idea is to use some Gronwall type integral inequalities. As an illustration, we present some examples with simulations to show the applicability of the obtained results.

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Non-linear integral inequalities and applications to asymptotic stability

Cited by 10 publications

References 0 publications

New results on the uniform exponential stability of nonautonomous perturbed dynamical systems

New results on the uniform exponential stability of nonautonomous perturbed dynamical systems

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

h-stability and boundedness results for solutions to certain nonlinear perturbed systems

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