Asymptotic limits and stabilization for the 1D nonlinear Mindlin-Timoshenko system

Abstract

This paper shows how the so called von Kármán model can be obtained as a singular limit of a modified Mindlin-Timoshenko system when the modulus of elasticity in shear k tends to infinity, provided a regularizing term through a fourth order dispersive operator is added. Introducing damping mechanisms, the authors also show that the energy of solutions for this modified Mindlin-Timoshenko system decays exponentially, uniformly with respect to the parameter k. As k → ∞, the authors obtain the damped von Kármán model with associated energy exponentially decaying to zero as well.

Exponential stabilization of the Timoshenko system by a thermal effect with an oscillating kernel

“…Finally in order to get the term −  Ω ξ 2 t dx we define the functional (33) and prove the following lemma.…”

A localized nonstandard stabilizer for the Timoshenko beam

“…This last observation may have led some authors to use two feedback controls to exponentially stabilize the Timoshenko system independently of (1); e.g. [3,6,11]. The present note fits in the latter framework, but the damping scheme involves the same feedback control in both equations unlike the cited works where two independent controls are employed.…”

General decay of the solution to a nonlinear viscoelastic modified von-Kármán system with delay

“…In [4], Araruna et al have showed how the so called von Kármán model (6) can be obtained as a singular limit of a modified Mindlin-Timoshenko system (4) when the modulus of elasticity in shear k tends to infinity, provided a regularizing term through a fourth order dispersive operator is added. Introducing damping mechanisms, the authors also show that the energy of solutions for this modified Mindlin-Timoshenko system decays exponentially, uniformly with respect to the parameter k. As k −→ ∞, the authors obtain the damped von Kármán model with associated energy exponentially decaying to zero as well.…”

“…Uniqueness can be proved by the straightforward methods and Gronwall's inequality. 4. General decay.…”