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Akram Ben Aissa

Publications and source records attributed to Akram Ben Aissa.

12 recordsLinked to original sources
arXiv2023

Stabilization of a locally transmission problems of two strongly-weakly coupled wave systems

In this paper, we embark on a captivating exploration of the stabilization of locally transmitted problems within the realm of two interconnected wave systems. To begin, we wield the formidable Arendt-Batty criteria\cite{AW} to affirm the resolute stability of our system. Then, with an artful fusion of a frequency domain approach and the multiplier method, we unveil the exquisite phenomenon of exponential stability, a phenomenon that manifests when the waves of the second system synchronize their propagation speeds. In cases where these speeds diverge, our investigation reveals a graceful decay of our system's energy, elegantly characterized by a polynomial decline at a rate of $t^{-1}$.

math.AP↗
arXiv2022

Global existence and Asymptotic behavior for a system of wave equation in presence of distributed delay term

In this paper, we consider the following viscoelastic coupled wave equation with a delay term: $$ \begin{gathered} u_{tt}(x,t)-Lu(x,t)-\int_0^t g_1(t-σ)L u(x,σ)dσ+ μ_{1}u_{t}(x,t) + \int_{τ_1}^{τ_2} μ_2(s)u_{t}(x,t-s)ds + f_1(u,\upsilon)=0, \\ \upsilon_{tt}(x,t) - L\upsilon(x,t) - \int_0^t g_{2}(t-σ)L \upsilon(x,σ)dσ+ μ_3\upsilon_t(x,t) + \int_{τ_1}^{τ_2} μ_4(s)\upsilon_{t}(x,t-s)ds + f_{2}(u,\upsilon)=0, \end{gathered} $$ in a bounded domain. Under appropriate conditions on $μ_{1}$, $μ_{2}$, $μ_{3}$ and $μ_{4}$, we prove global existence result by combining the energy method with the Faedo-Galerkin's procedure. In addition , we focus on asymptotic behavior by using an appropriate Lyapunov functional.

math.AP↗
arXiv2021

Well-posedness and Stabiliy result of Petrovsky equation with a nonlinear strong damping and delay term

In this paper we consider a nonlinear Petrovsky equation in a bounded domain with a delay term and a strong dissipation \begin{align*} u_{tt} + Δ^{2} u -μ_1g_1( Δ( u_t(x,t))) -μ_2g_2( Δ(u_t(x,t-τ))) =0. \end{align*} We prove the existence of global solutions in suitable Sobolev spaces by using the energy method combined with Faedo-Galarkin method under condition on the weight of the delay term in the feedback and the weight of the term without delay. Furthermore, we study general stability estimates by using some properties of convex functions.

math.AP↗
arXiv2021

Well-posedness and direct internal stability of coupled non-degenrate Kirchhoff system via heat conduction

In the paper under study, we consider the following coupled non-degenerate Kirchhoff system \begin{equation}\label{P} \left \{ \begin{aligned} &\displaystyle y_{tt}-\upvarphi\Big(\int_Ω| \nabla y |^2\,dx\Big)Δy +\upalpha Δ\uptheta=0, &\mbox{ in }&\; Ω\times (0, +\infty)\\ &\displaystyle \uptheta_t-Δ\uptheta-\upbeta Δy_t =0, &\mbox{ in }&\; Ω\times (0, +\infty)\\ &\displaystyle y=\uptheta=0,\; &\mbox{ on }&\;\partialΩ\times(0, +\infty)\\ %&\displaystyle y=0,\; &\mbox{ on }&\;\partialΩ\times(0, +\infty)\\ %&\displaystyle \partial_νy=0, &\mbox{ on }&\;Γ_1\times(0, +\infty)\\ &\displaystyle y(\cdot, 0)=y_0, \; y_t(\cdot, 0)=y_1,\;\uptheta(\cdot, 0)=\uptheta_0, \; \; &\mbox{ in }&\; Ω\\ \end{aligned} \right. \end{equation} where $Ω$ is a bounded open subset of $\mathbb{R}^n$, $\upalpha$ and $\upbeta$ be two nonzero real numbers with the same sign and $\upvarphi$ is given by $\upvarphi(s)= \mathfrak{m}_0+\mathfrak{m}_1s$ with some positive constants $\mathfrak{m}_0$ and $\mathfrak{m}_1$. So we prove existence of solution and establish its exponential decay. The method used is based on multiplier technique and some integral inequalities due to Haraux and Komornik\cite{H1,KOM}.

math.AP↗
arXiv2020

Well-posedness and exponential decay for the Euler-Bernoulli beam conveying fluid equation with non-constant velocity and dynamical boundary conditions

In this paper, we consider an Euler-Bernoulli beam equation with time-varying internal fluid. We assume that the fluid is moving with non-constant velocity and dynamical boundary conditions are satisfied. We prove the existence and uniqueness of global solution under suitable assumptions on the tension of beam and on the parameters of the problem. Afterwards, we establish the exponential stability of the solution by introducing a suitable Lyapunov functional.

math.AP↗
arXiv2018

A frequency approach for stabilization of one-dimensional degenerate wave equation

In this paper, we are concerned with the study of stabilization problem for the following strongly degenerate wave equation in one space dimension $$w_{tt}(x,t)-\left(x^αw_x(x,t)\right)_x=0$$ where ${\bfα\in [1,2)}$. Thus, using a frequency domain method inspired from \cite{BT}, we prove the polynomial decays of its total energy with $t^{-\nicefrac{1}{2}}$ decay rate.

math.AP↗
arXiv2018

Stability results for viscoelastic wave equation with dynamic boundary conditions

In this paper we consider wave viscoelastic equation with dynamic boundary condition in a bounded domain, we establish a general decay result of energy by exploiting the frequency domain method which consists in combining a contradiction argument and a special analysis for the resolvent of the operator of interest with assumptions on past history relaxation function.

math.AP↗
arXiv2013

Weak controllability of second order evolution systems and applications

Controllability and observability are important properties of a distributed paramater systems.The equivalence between the notion of exact observability and exact controllability holds in general. In this work, we define a new notion of controllability say weak which is related to some weak observability inequality and we give the equivalence between.

math.OC↗
arXiv2013

Grushin problems and control theory: Formulation and examples

In this paper we give a new formulation of an abstract control problem in terms of a Grushin problem, so that we will reformulate all notions of controllability, observability and stability in a new form that gives readers an easy interpretation of these notions.

math.OC↗