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Farhat Shel

Publications and source records attributed to Farhat Shel.

10 recordsLinked to original sources
arXiv2024

Well-posedness and stability of abstract thermoelastic delayed systems

In this paper, we consider a stabilization problem of a generalized thermoelastic system (the so called $α-β$ system) with delay in a part of the coupled system. For each case, we prove the well-posedness of the corresponding system using semigroup approach, then under some sufficient conditions we establish some results of exponential and polynomial stability of the system through a frequency-domain approach. The results are applied to concrete examples in thermoelasticity.

math.AP↗
arXiv2022

A note on exponential stability of a thermoelastic system with internal delay

The presence of a delay in a thermoelastic system destroys the well-posedness and the stabilizing effect of the heat conduction. To avoid this problem we add to the system, at the delayed equation, a Kelvin-Voigt damping. In this note we point on the exponential stability of such system in order to improve the mean result in our paper Well-posedness and exponential stability of a thermoelastic system with internal delay (Applicable Analysis J 101, 4851-4865, 2022). We use a frequency domain method in the proof of stability.

math.AP↗
arXiv2022

Note on stability of an abstract coupled hyperbolic-parabolic system: singular case

In this paper we try to complete the stability analysis for an abstract system of coupled hyperbolic and parabolic equations $$ \left\{ \begin{array}{lll} \ds u_{tt} + Au - A^αw = 0, \\ w_t + A^αu_t + A^βw = 0,\\ u(0) = u_0, u_t(0) = u_1, w(0) = w_0, \end{array} \right. $$ where $A$ is a self-adjoint, positive definite operator on a complex Hilbert space $H$, and $(α, β) \in [0,1] \times [0,1]$, which is considered in \cite{Amk}, and after, in \cite{liu1}. Our contribution is to identify a fine scale of polynomial stability of the solution in the region $ S_3: = \left\{(α,β) \in [0,1] \times [0,1]; \, β< 2α-1 \right\}$ taking into account the presence of a singularity at zero.

math.AP↗
arXiv2022

Second sound thermoelastic stability of a string/beam structure

In this paper we study the one dimensional thermoelastic transmission problem in a special string/beam structure: the two components are coupled at an interface (identified to $0$). Either the string or the beam is supposed thermoelastic, the heat flux is given by the Cattaneo's law instead of the usual Fourier's law. We prove that the the energy decay of the whole system is exponential if the string is thermoelastic. When only the beam is thermoelastic, we prove that the energy of the coupling string/beam decays polynomially to zero as $\frac{1}{t}$ and the decay rate can be, at most, polynomially stable of order $\frac{1}{t^2}$.

math.AP↗
arXiv2021

Regularity of the semigroups associated with some damped coupled elastic systems II: a nondegenerate fractional damping case

In this paper, we examine regularity issues for two damped abstract elastic systems; the damping and coupling involve fractional powers $μ, θ$, with $0 \leq μ, θ\leq 1$, of the principal operators. The matrix defining the coupling and damping is nondegenerate. This new work is a sequel to the degenerate case that we discussed recently in \cite{kfl}. First, we prove that for $1/2 \leq μ, θ\leq 1$, the underlying semigroup is analytic. Next, we show that for $\min(μ,θ) \in (0,1/2)$, the semigroup is of certain Gevrey classes. Finally, some examples of application are provided.

math.AP↗
arXiv2020

Well-posedness and exponential stability of a thermoelastic system with internal delay

The presence of a delay in a thermoelastic system destroys the well-posedness and the stabilizing effect of the heat conduction [17]. To avoid this problem we add to the system, at the delayed equation, a Kelvin-Voigt damping. At first, we prove the well-posedness of the system by the semigroup theory. Next, under appropriate assumptions, we prove the exponential stability of the system by introducing a suitable Lyapunov functional.

math.AP↗
arXiv2018

Stability of the wave equations on a tree with local Kelvin-Voigt damping

In this paper we study the stability problem of a tree of elastic strings with local Kelvin-Voigt damping on some of the edges. Under the compatibility condition of displacement and strain and continuity condition of damping coefficients at the vertices of the tree, exponential/polynomial stability are proved.

math.AP↗
arXiv2015

Feedback stabilization of a simplified model of fluid-structure interaction on a tree

In this paper we study the dynamic feedback stability for a simplified model of fluid-structure interaction on a tree. We prove that, under some conditions, the energy of the solutions of the system decay exponentially to zero when the time tends to infinity. Our technique is based on a frequency domain method and a special analysis for the resolvent.

math.AP↗
arXiv2015

Stability of a tree-shaped network of strings and beams

In this paper we study the stability of a tree-shaped network of elastic strings and beams with some feedbacks at the ends. The whole system is asymptotically stable. Moreover, the energy of the solution decay exponentially to zero if there is no beam following a string ( from the root to the leaves) and decay polynomially if not. We use a frequency domain method from the semigroups theory.

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