arXiv subjects

Chiraz Kassem

Publications and source records attributed to Chiraz Kassem.

4 recordsLinked to original sources

arXiv2023

Stability for an interface transmission problem of wave-plate equations with dynamical boundary controls

We investigate a two-dimensional transmission model consisting of a wave equation and a Kirchhoff plate equation with dynamical boundary controls under geometric conditions. The two equations are coupled through transmission conditions along a steady interface between the domains in which the wave and plate equations evolve, respectively. Our primary concern is the stability analysis of the system, which has not appeared in the literature. For this aim, using a unique continuation theorem, the strong stability of the system is proved without any geometric condition and in the absence of compactness of the resolvent. Then, we show that our system lacks exponential (uniform) stability. However, we establish a polynomial energy decay estimate of type 1/t for smooth initial data using the frequency domain approach from semigroup theory, which combines a contradiction argument with the multiplier technique. This method leads to certain geometrical conditions concerning the wave's and the plate's domains.

math.AP↗

arXiv2022

Polynomial stabilization of non-smooth direct/indirect elastic/viscoelastic damping problem involving Bresse system

We consider an elastic/viscoelastic transmission problem for the Bresse system with fully Dirichlet or Dirichlet-Neumann-Neumann boundary conditions. The physical model consists of three wave equations coupled in certain pattern. The system is damped directly or indirectly by global or local Kelvin-Voigt damping. Actually, the number of the dampings, their nature of distribution (locally or globally) and the smoothness of the damping coefficient at the interface play a crucial role in the type of the stabilization of the corresponding semigroup. Indeed, using frequency domain approach combined with multiplier techniques and the construction of a new multiplier function, we establish different types of energy decay rate (see the table of stability results below). Our results generalize and improve many earlier ones in the literature and in particular some studies done on the Timoshenko system with Kelvin-Voigt damping.

math.AP↗

arXiv2021

Exact controllability and stabilization of locally coupled wave equations : theoretical results

In this paper, we study the exact controllability and stabilization of a system of two wave equations coupled by velocities with an internal, local control acting on only one equation. We distinguish two cases. In the first one, when the waves propagate at the same speed: using a frequency domain approach combined with multiplier technique, we prove that the system is exponentially stable when the coupling region satisfies the geometric control condition GCC. Following a result of Haraux ([11]), we establish the main indirect observability inequality. This results leads, by the HUM method, to prove that the total system is exactly controllable by means of locally distributed control. In the second case, when the waves propagate at different speed, we establish an exponential decay rate in the weak energy space. Consequently, the system is exactly controllable using a result of [11].

math.AP↗

arXiv2021

Numerical study of the stabilization of 1D locally coupled wave equations

In this paper, we study the numerical stabilization of a 1D system of two wave equations coupled by velocities with an internal, local control acting on only one equation. In the theoretical part of this study, we distinguished two cases. In the first one, the two waves assumed propagate at the same speed. Under appropriate geometric conditions, we had proved that the energy decays exponentially. While in the second case, when the waves propagate at different speeds, under appropriate geometric conditions, we had proved that the energy decays only at a polynomial rate. In this paper, we confirmed these two results in a 1D numerical approximation. However, when the coupling region does not intersect the damping region, the stabilization of the system is still theoretically an open problem. But, here in both cases, we observed an unpredicted behavior : the energy decays at an exponential rate when the propagation speeds are the same or at a polynomial rate when they are different.

math.AP↗