On the stability of Bresse system with one discontinuous local internal Kelvin–Voigt damping on the axial force

The influence of the physical coefficients of a Bresse system with one singular local viscous damping in the longitudinal displacement on its stabilization

“…Here ρ 1 = ρA, ρ 2 = ρI, k 1 = kGA, k 3 = EA, k 2 = EI and l = R −1 , in which ρ is the density of the material, E the modulus of the elasticity, G the shear modulus, k the shear factor, A the cross-sectional area, I the second moment of area of the cross section, R the radius of the curvature, and l the curvature. There are several publications concerning the stabilization of Bresse system with frictional or another kinds of damping (see [1], [2], [3], [7], [8], [15], [16], [17], [18], [19], [21], [20], [22], [30], [29], [31], [32], [35], [38] and [28]). We note that by neglecting w (l → 0) in (1.5), the Bresse system reduces to the following conservative Timoshenko system:…”

“…Let (φ 1 , φ 2 , φ 3 ) ∈ H 1 0 (0, L) 3 . Multiplying (2.8), (2.10) and (2.14) by φ 1 , φ 2 and φ 3 respectively, integrating over (0, L), then using formal integrations by parts, we obtain…”

“…3 and L is an antilinear and continuous form on H 1 0 (0, L) 3 . Then, it follows by Lax-Milgram theorem that (2.15) admits a unique solution (v 1 , v 3 , v 5 ) ∈ H 1 0 (0, L) 3 .…”

“…3 and L is an antilinear and continuous form on H 1 0 (0, L) 3 . Then, it follows by Lax-Milgram theorem that (2.15) admits a unique solution (v 1 , v 3 , v 5 ) ∈ H 1 0 (0, L) 3 . By taking test-functions (φ 1 , φ 2 , φ 3 ) ∈ (D(0, L)) 3 , we see that ((2.8),…”

“…3 = O(|λ| −1 ), v 5 = O(|λ| −1 ), f 5 = o(1), f 5x = o(1) and f 6 = o(1), we obtain L 63), (4.64), (4.65) and using the fact that ∈ {0, 2, 4}, then using integration by parts, we obtain (4.62).Step In this step, we conclude the proof of Lemma 4.7. For this aim, take h = xg 1 + (x − L)g 2 in (4.62), we obtainα1 |λv 5 | 2 dx.Now, take α 1 = 4ε and α 2 = β − 4ε in the above equation, then using Lemmas 4.1, 4.2, 4.4, when (H 1 ) holds, and (4.45), we obtain (4.59).…”