arXiv subjects

Abdelmouhcene Sengouga

Publications and source records attributed to Abdelmouhcene Sengouga.

5 recordsLinked to original sources

arXiv2023

Boundary stabilization of a vibrating string with variable length

We study small vibrations of a string with time-dependent length $\ell(t)$ and boundary damping. The vibrations are described by a 1-d wave equation in an interval with one moving endpoint at a speed $\ell'(t)$ slower than the speed of propagation of the wave c=1. With no damping, the energy of the solution decays if the interval is expanding and increases if the interval is shrinking. The energy decays faster when the interval is expanding and a constant damping is applied at the moving end. However, to ensure the energy decay in a shrinking interval, the damping factor $η$ must be close enough to the optimal value $η=1$, corresponding to the transparent condition. In all cases, we establish lower and upper estimates for the energy with explicit constants.

math.AP↗

arXiv2022

Energy decay estimates for an axially travelling string damped at one end

We study the small vibrations of an axially travelling string with a dashpoint damping at one end. The string is modelled by a wave equation in a time-dependent interval with two endpoints moving at a constant speed $v$. For the undamped case, we obtain a conserved functional equivalent to the energy of the solution. We derive precise upper and lower estimates for the exponential decay of the energy with explicit constants. These estimates do not seem to be reported in the literature even for the non-travelling case $v=0$.

math.AP↗

arXiv2022

Free vibrations of axially moving strings: Energy estimates and boundary observability

We study the small vibrations of axially moving strings described by a wave equation in an interval with two endpoints moving in the same direction with a constant speed. The solution is expressed by a series formula where the coefficients are explicitly computed in function of the initial data. We also define an energy expression for the solution that is conserved in time. Then, we establish boundary observability inequalities with explicit constants.

math.AP↗

arXiv2018

Exact Boundary Observability and Controllability of the Wave Equation in an Interval with two Moving Endpoints

We study the wave equation in an interval with two linearly moving endpoints. We give the exact solution by a series formula, then we show that the energy of the solution decay at the rate $1/t$. We also establish observability results, at one or two endpoints, in a sharp time. Moreover, using the Hilbert uniqueness method, we derive exact boundary controllability results.

math.AP↗

arXiv2017

Observability and controllability of the 1--d wave equation in domains with moving boundary

By mean of generalized Fourier series and Parseval's equality in weighted $L^{2}$--spaces, we derive a sharp energy estimate for the wave equation in a bounded interval with a moving endpoint. Then, we show the observability, in a sharp time, at each of the endpoints of the interval. The observability constants are explicitly given. Using the Hilbert Uniqueness Method we deduce the exact boundary controllability of the wave equation.

math.FA↗