Abstract
We consider the Bresse system with three control boundary conditions of fractional derivative type. We prove the polynomial decay result with an estimation of the decay rates. Our result is established using the semigroup theory of linear operators and a result obtained by Borichev and Tomilov.
Highlights
In this paper we discuss the existence and energy decay rate of solutions for the initial boundary value problem of the linear Bresse system of the type ρ1φtt − Gh(φx + ψ + lω)x − lEh(ωx − lφ) = 0 (P )ρ2ψtt − EIψxx + Gh(φx + ψ + lω) = 0ρ1ωtt − Eh(ωx − lφ)x + lGh(φx + ψ + lω) = 0 where (x, t) ∈ (0, L) × (0, +∞)
Ρ1ωtt − Eh(ωx − lφ)x + lGh(φx + ψ + lω) = 0 where (x, t) ∈ (0, L) × (0, +∞). This system is subject to the boundary conditions of the form φ(0, t) = 0, ψ(0, t) = 0, ω(0, t) = 0 Gh(φx + ψ + lω)(L, t) = −γ1∂tα,ηφ(L, t) EIψx(L, t) = −γ2∂tα,ηψ(L, t) Eh(ωx − lφ)(L, t) = −γ3∂tα,ηω(L, t) in (0, +∞) in (0, +∞) in (0, +∞) in (0, +∞)
Case 2 η = 0: We aim to show that an infinite number of eigenvalues of A approach the imaginary axis which prevents the Bresse system (P ) from being exponentially stable
Summary
In this paper we discuss the existence and energy decay rate of solutions for the initial boundary value problem of the linear Bresse system of the type ρ1φtt − Gh(φx + ψ + lω)x − lEh(ωx − lφ) = 0. Raposo et al [20] proved the exponential decay of the solution for the following linear system of Timoshenko-type beam equations with linear frictional dissipative terms: ρ1φtt − Gh(φx + ψ)x + μ1φt = 0 ρ2ψtt − EIψxx + Gh(φx + ψ) + μ1ψt = 0. We will prove that the operator λI −A is surjective for λ > 0 For this purpose, let (f1, f2, f3, f4, f5, f6, f7, f8, f9)T ∈ H, we seek U = (φ, u, φ1, ψ, v, φ2, ω, ω, φ3)T ∈ D(A) solution of the following system of equations λφ f1, λλuφ1−+Gρ(1ξh2(φ+xη+)φψ. The semigroup {S(t)}t≥0 is asymptotically stable, i.e, S(t)z H → 0 as t → ∞ for any z ∈ H
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