Papers by Mohamed Hammami
Mathematica, May 15, 2024
In this paper, we give a new integral inequality which is used to study the asymptotic behavior o... more In this paper, we give a new integral inequality which is used to study the asymptotic behavior of solutions for a class of nonlinear dynamic systems with small perturbation using a numerical approach. We provide some new results on the stability of perturbed systems where necessary and sufficient condition is derived. We show that the perturbed nonlinear system can be globally uniformly practically asymptotically stable provided that the bound of perturbation is small enough. A numerical example is presented to illustrate the validity of the main result. MSC 2020. 30C45.
Applied Mathematics and Optimization, Feb 9, 2021
The approach of Lyapunov functions is one of the most efficient ones for the investigation of the... more The approach of Lyapunov functions is one of the most efficient ones for the investigation of the stability of stochastic systems, in particular, of singular stochastic systems. The main objective of the paper is the analysis of the stability of stochastic perturbed singular systems by using Lyapunov techniques under the assumption that the initial conditions are consistent. The uniform exponential stability in mean square and the practical uniform exponential stability in mean square of solutions of stochastic perturbed singular systems based on Lyapunov techniques are investigated. Moreover, we study the problem of stability and stabilization of some classes of stochastic singular systems. Finally, an illustrative example is given to illustrate the effectiveness of the proposed results.
Journal of Linear and Topological Algebra, Dec 1, 2016
In this paper, global uniform exponential stability of perturbed dynamical systems is studied by ... more In this paper, global uniform exponential stability of perturbed dynamical systems is studied by using Lyapunov techniques. The system presents a perturbation term which is bounded by an integrable function with the assumption that the nominal system is globally uniformly exponentially stable. Some examples in dimensional two are given to illustrate the applicability of the main results.
Journal of Engineering Mathematics, Aug 28, 2021
This paper is concerned with the almost sure partial practical stability of stochastic differenti... more 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.
Research Square (Research Square), May 8, 2023
In this paper we investigate the practical partial asymptotic and exponential stability of time-v... more In this paper we investigate the practical partial asymptotic and exponential stability of time-varying nonlinear systems. We derive some sufficient conditions that guarantee practical partial stability of a large class of perturbed systems using Lyapunov's theory where a converse theorem is presented. Therefore, we generalized some works which are already made in the literature. Furthermore, some examples are presented.
Archives of Control Sciences, Mar 1, 2013
In this paper, we study the observer design problem for a class of nonlinear systems. Specificall... more In this paper, we study the observer design problem for a class of nonlinear systems. Specifically, we design an exponential observer for a separately excited DC motor. Moreover, a stabilizing controller is designed for the system to ensure the exponential stability of the solutions toward their desired values. Simulations results show that proposed observer is able to reconstruct the states of the system. In addition, the simulation results indicate that the designed controller works well.
Turkish Journal of Mathematics, 2016
We consider the nonlinear equation −∆u = |u| p−1 u−εu in Ω, u = 0 on ∂Ω, where Ω is a smooth boun... more We consider the nonlinear equation −∆u = |u| p−1 u−εu in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in R n , n ≥ 4 , ε is a small positive parameter, and p = (n + 2)/(n − 2). We study the existence of sign-changing solutions that concentrate at some points of the domain. We prove that this problem has no solutions with one positive and one negative bubble. Furthermore, for a family of solutions with exactly two positive bubbles and one negative bubble, we prove that the limits of the blow-up points satisfy a certain condition.
Journal of Applied Analysis, 2008
In this paper, we study the stabilization problem of uncertain systems. We treat a class of uncer... more In this paper, we study the stabilization problem of uncertain systems. We treat a class of uncertain systems whose nominal part is affine in the control and whose uncertain part is bounded by a known affine function of the control, when the control is bounded by a specified constant.
Systems & Control Letters, Nov 1, 2017
This paper is devoted to the investigation of the practical exponential stability of impulsive st... more This paper is devoted to the investigation of the practical exponential stability of impulsive stochastic functional differential equations. The main tool used to prove the results is the Lyapunov-Razumikhin method which has proven very useful in dealing with stability problems for differential systems when the delays involved in the equations are not differentiable but only continuous. An illustrative example is also analyzed to show the applicability and interest of the main results.
Stochastics An International Journal of Probability and Stochastic Processes, Jun 4, 2015
The method of Lyapunov functions is one of the most effective ones for the investigation of stabi... more The method of Lyapunov functions is one of the most effective ones for the investigation of stability of dynamical systems, in particular, of stochastic differential systems. The main purpose of the paper is the analysis of the stability of stochastic differential equations by using Lyapunov functions when the origin is not necessarily an equilibrium point. The global uniform boundedness and the global practical uniform exponential stability of solutions of stochastic differential equations based on Lyapunov techniques are investigated. Furthermore, an example is given to illustrate the applicability of the main result.
International Journal of Robust and Nonlinear Control, Apr 26, 2021
In this paper, we investigate the asymptotic behaviors of the solutions of nonlinear dynamic syst... more In this paper, 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 terms, the equilibrium point remains (globally) uniformly exponentially stable. The results we obtained 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 are presented in order to illustrate the validity of our study, especially when the standard Lyapunov approaches are useless.
Complexity, Apr 14, 2022
Fractional calculus is nowadays an efficient tool in modelling many interesting nonlinear phenome... more Fractional calculus is nowadays an efficient tool in modelling many interesting nonlinear phenomena. is study investigates, in a novel way, the Ulam-Hyers (HU) and Ulam-Hyers-Rassias (HUR) stability of differential equations with general conformable derivative (GCD). In our analysis, we employ some version of Banach fixed-point theory (FPT). In this way, we generalize several earlier interesting results. Two examples are given at the end to illustrate our results.
New Zealand Journal of Mathematics
This paper treats the concept of practical uniform $h$-stability for such perturbed dynamical sys... more This paper treats the concept of practical uniform $h$-stability for such perturbed dynamical systems as an extension of practical uniform exponential stability. We present a converse Lyapunov theorem and we give sufficient conditions that guarantee practical uniform $h$-stability for a time-varying perturbed system using the Gronwall-Bellman inequality and Lyapunov's theory. Some examples are introduced to illustrate the applicability of the main results.
Archives of Control Sciences
Many nonlinear dynamical systems can present a challenge for the stability analysis in particular... more Many nonlinear dynamical systems can present a challenge for the stability analysis in particular the estimation of the region of attraction of an equilibrium point. The usual method is based on Lyapunov techniques. For the validity of the analysis it should be supposed that the initial conditions lie in the domain of attraction. In this paper, we investigate such problem for a class of dynamical systems where the origin is not necessarily an equilibrium point. In this case, a small compact neighborhood of the origin can be estimated as an attractor for the system. We give a method to estimate the basin of attraction based on the construction of a suitable Lyapunov function. Furthermore, an application to Lorenz system is given to verify the effectiveness of the proposed method.
Archives of Control Sciences
An important application of state estimation is feedback control: an estimate of the state (typic... more An important application of state estimation is feedback control: an estimate of the state (typically the mean estimate) is used as input to a state-feedback controller. This scheme is known as observer based control, and it is a common way of designing an output-feedback controller (i.e. a controller that does not have access to perfect state measurements). In this paper, under the fact that both the estimator dynamics and the state feedback dynamics are stable we propose a separation principle for Takagi-Sugeno fuzzy control systems with Lipschitz nonlinearities. The considered nonlinearities are Lipschitz or meets an integrability condition which have no influence on the LMI to prove the stability of the associated closed-loop system. Furthermore, we give an example to ullistrate the applicability of the main result.
Mathematica Moravica, 2022
In this survey, we introduce the notion of stability of time varying nonlinear systems. In partic... more In this survey, we introduce the notion of stability of time varying nonlinear systems. In particular we investigate the notion of global practical exponential stability for non-autonomous systems. The proposed approach for stability analysis is based on the determination of the bounds of perturbations that characterize the asymptotic convergence of the solutions to a closed ball centered at the origin.
Electronic Journal of Qualitative Theory of Differential Equations
This paper stands for the almost sure practical stability of nonlinear stochastic differential de... more This paper stands for the almost sure practical stability of nonlinear stochastic differential delay equations (SDDEs) with a general decay rate. We establish some sufficient conditions based upon the construction of appropriate Lyapunov functionals. Furthermore, we provide some numerical examples to validate the effectiveness of the abstract results of this paper.
arXiv (Cornell University), Aug 25, 2004
In this paper we consider the following biharmonic equation with critical exponent (Pε) : ∆ 2 u =... more In this paper we consider the following biharmonic equation with critical exponent (Pε) : ∆ 2 u = Ku n+4 n−4 −ε , u > 0 in Ω and u = ∆u = 0 on ∂Ω, where Ω is a smooth bounded domain in R n , n ≥ 5, ε is a small positive parameter, and K is a smooth positive function in Ω. We construct solutions of (Pε) which blow up and concentrate at strict local maximum of K either at the boundary or in the interior of Ω. We also construct solutions of (Pε) concentrating at an interior strict local minimum point of K. Finally, we prove a nonexistence result for the correponding supercritical problem which is in sharp contrast to what happened for (Pε).
arXiv: Dynamical Systems, 2019
In this paper, we give a new integral inequalities which are used to study the asymptotic behavio... more In this paper, we give a new integral inequalities which are used to study the asymptotic behavior of solutions of nonlinear dynamic systems with small perturbation. We derive some new results on the stability of nonlinear systems with small perturbation. Explicitly, we derive a necessary and sufficient condition for a linear system with small perturbation to be globally uniformly exponentially stable at the origin. We show that the nonlinear system with small perturbation can be globally uniformly practically asymptotically stable provided that the bound of perturbation is small enough. A numerical example is presented to illustrate the validity of the main result.
Journal of Engineering Mathematics, 2021
This paper is concerned with the almost sure partial practical stability of stochastic differenti... more 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.
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Papers by Mohamed Hammami