Stability to weak dissipative Bresse system

Abstract

In this paper we study the Bresse system with frictional dissipation working only on the angle displacement. Our main result is to prove that this dissipative mechanism is enough to stabilize exponentially the whole system provided the velocities of waves propagations are the same. This result is significative only from the mathematical point of view since in practice the velocities of waves propagations are always different. In that direction we show that when the velocities are not the same, the system is not exponentially stable and we prove that the solution in this case goes to zero polynomially, with rates that can be improved by taking more regular initial data. Finally, we give some numerical result to verify our analytical results.

Well-Posedness and Asymptotic Stability of Solutions to the Bresse System under Cattaneo's Law with Infinite Memories and Time-Varying Delays

“…The Bresse system is known as the circular arch problem (see [16] for details) and is given by the following equations: 1) where N = k 3 (w x − Iϕ), Q = k 1 (ϕ x + Iw + ψ), M = k 2 ψ x denote the axial force, the shear force and the bending moment, respectively, ρ 1 = ρA, ρ 2 = ρI, k 3 = EA, k 1 = k GA, k 2 = EI, I = R −1 . The functions w, ϕ and ψ are the longitudinal, vertical and shear angle displacements, respectively.…”

“…During the last few decades, there are many works treating about existence and stabilization of Bresse system. Alabau Boussouira et al [1] considered a Bresse system with one frictional damping working only on the angle displacement. The authors proved that the exponential decay exists when the velocities of the wave propagations are the same.…”

Long-time dynamics of two classes of beam and plate equations

“…2) We observe that when the curvature → 0 the system (1.2) uncouples into the well-known Timoshenko system,    ρ 1 ϕ tt − k(ϕ x + ψ) x = 0 ρ 2 ψ tt − bψ xx + k(ϕ x + ψ) = 0 (1. 3) and an independent wave equation ρ 1 w tt − k 0 w xx = 0.…”

“…Concerning to the model with only one weak damping on the angle displacement (ψ t ), the result in [2] asserts that the system (1.2) is exponentially stable only when the mathematical condition (1.4) holds. When this equality fails, it is shown in [2] that system with mixed boundary condition (Dirichlet-Neumann-Dirichlet) does not have exponential decay rate.…”

“…When this equality fails, it is shown in [2] that system with mixed boundary condition (Dirichlet-Neumann-Dirichlet) does not have exponential decay rate. Instead, the solution goes to zero polynomially with rate depending on the coefficients.…”