arXiv subjects

Fadhel Al-Musallam

Publications and source records attributed to Fadhel Al-Musallam.

3 recordsLinked to original sources

arXiv2017

Asymptotic analysis of a 2D overhead crane with input delays in the boundary control

The paper investigates the asymptotic behavior of a 2D overhead crane with input delays in the boundary control. A linear boundary control is proposed. The main feature of such a control lies in the facts that it solely depends on the velocity but under the presence of time-delays. We end-up with a closed-loop system where no displacement term is involved. It is shown that the problem is well-posed in the sense of semigroups theory. LaSalle's invariance principle is invoked in order to establish the asymptotic convergence for the solutions of the system to a stationary position which depends on the initial data. Using a resolvent method it is proved that the convergence is indeed polynomial.

math.AP↗

arXiv2016

The equivalent medium for the elastic scattering by many small rigid bodies and applications

We deal with the elastic scattering by a large number $M$ of rigid bodies, $D_m:=εB_m+z_m$, of arbitrary shapes with $ 0<\textcolor{black}ε<<1$ and with constant Lamé coefficients $λ$ and $μ$. We show that, when these rigid bodies are distributed arbitrarily (not necessarily periodically) in a bounded region $Ω$ of $\mathbb{R}^3$ where their number is $M:=M(\textcolor{black}ε):=O(\textcolor{black}ε^{-1})$ and the minimum distance between them is $d:=d(\textcolor{black}ε)\approx \textcolor{black}ε^{t}$ with $t$ in some appropriate range, as $\textcolor{black}ε \rightarrow 0$, the generated far-field patterns approximate the far-field patterns generated by an equivalent medium given by $ω^2ρI_3-(K+1)\mathbf{C}_0 $ where $ρ$ is the density of the background medium (with $I_3$ as the unit matrix) and $(K+1)\mathbf{C}_0$ is the shifting (and possibly variable) coefficient. This shifting coefficient is described by the two coefficients $K$ and $\mathbf{C}_0$ (which have supports in $\overlineΩ$) modeling the local distribution of the small bodies and their geometries, respectively. In particular, if the distributed bodies have a uniform spherical shape then the equivalent medium is isotropic while for general shapes it might be anisotropic (i.e. $\mathbf{C}_0$ might be a matrix). In addition, if the background density $ρ$ is variable in $Ω$ and $ρ=1$ in $\mathbb{R}^3\setminus{\overlineΩ}$, then if we remove from $Ω$ appropriately distributed small bodies then the equivalent medium will be equal to $ω^2 I_3$ in $\mathbb{R}^3$, i.e. the obstacle $Ω$ characterized by $ρ$ is approximately cloaked at the given and fixed frequency $ω$.

math.AP↗

arXiv2014

Location and size estimation of small rigid bodies using elastic far-fields

We are concerned with the linearized, isotropic and homogeneous elastic scattering problem by (possibly many) small rigid obstacles of arbitrary Lipschitz regular shapes in 3D. Based on the Foldy-Lax approximation, valid under a sufficient condition on the number of the obstacles, the size and the minimum distance between them, we show that any of the two body waves, namely the pressure waves P or the shear waves S, is enough for solving the inverse problem of detecting these scatterers and estimating their sizes. Further, it is also shown that the shear-horizontal part SH or the shear vertical part SV of the shear waves S are also enough for the location detection and the size estimation. Under some extra assumption on the scatterers, as the convexity assumption, we derive finer size estimates as the radius of the largest ball contained in each scatterer and the one of the smallest ball containing it. The two estimates measure, respectively, the thickness and length of each obstacle.

math.AP↗