On the stability of Mindlin–Timoshenko plates

Abstract

Abstract. We consider a Mindlin-Timoshenko model with frictional dissipations acting on the equations for the rotation angles. We prove that this system is not exponentially stable independent of any relations between the constants of the system, which is different from the analogous one-dimensional case. Moreover, we show that the solution decays polynomially to zero, with rates that can be improved depending on the regularity of the initial data.

On stability of hyperbolic thermoelastic Reissner–Mindlin–Timoshenko plates

“…In [7], Fernández Sare studied a linear Reissner-Mindlin-Timoshenko plate with a damping for both angle components…”

Stabilization of 2-d Mindlin-Timoshenko plates with localized acoustic boundary feedback

“…As a class of more comprehensive distributed parameter systems (Mindlin-Timoshenko Plate), the model contains shear effects in addition to displacement and rotational inertia effects, which reflects the vibration characteristics of thin plates in general case more accurately. Therefore, there is an extensive literature (such as Lagnese [20], Sare [34], Dalsen [11] and Messaoudi [25]) on the well-posedness and the asymptotic stabilization of Mindlin-Timoshenko beam or plate systems. The conservative two dimensional model is described as follows ψ tt , ϕ tt , ω tt − ρ 1 1 , ρ 1 2 , ρ 2 3 = 0, in Ω × (0, +∞), (1) where Ω ⊂ R 2 is an open bounded set and 0 denotes a three dimensional zero column vector.…”

“…(4) Provided that the nonlinear function p (x, y, ψ t , ϕ t , ω t ) satisfying the so-called "dissipativity assumptions" [11], the exponential stability of the energy of system (4) was obtained by means of Nakao's lemma [28]. However, this model is polynomially stable rather than exponentially stable, proved by Sare [34] who mainly considered putting the terms with frictional dissipation effects d 1 ψ t , d 2 ϕ t into rotation angle equations of this model with Dirichlet boundary conditions, namely,…”

“…As far as we know these estimates were not given in the literature. Compared with other related works (see [6,10,16,24,34,35,37]), our main contributions are summarized as follows:…”

“…As the functional a([ψ, ϕ, ω], [ψ,φ,ώ]) defined in the identity(9) with the propertya([ψ, ϕ, ω], [ψ,φ,ώ]) = a([ψ,φ,ω], [ψ,φ,ω]),then by using (71) together with the definition of the operator A in(34) and the formula (11), we geta([ψ, ϕ, ω], [ψ,φ,ώ]) = − Ω ψ 1 [ψ,φ,ω] + ϕ 2 [ψ,φ,ω] + ω 3 [ψ,φ,ω] dxdy…”