Timoshenko systems with indefinite damping

By J. E. Muñoz Rivera, Reinhard Racke. 2008

140 citations recorded by the source corpus.

Abstract We consider the Timoshenko system in a bounded domain ( 0 , L ) ⊂ R 1 . The system has an indefinite damping mechanism, i.e. with a damping function a = a ( x ) possibly changing sign, present only in the equation for the rotation angle. We shall prove that the system is still exponentially stable under the same conditions as in the positive constant damping case, and provided a ¯ = ∫ 0 L a ( x ) d x > 0 and ‖ a − a ¯ ‖ L 2 ϵ , for ϵ small enough. The decay rate will be described explicitly. In the arguments, we shall also give a new proof of exponential stability for the constant case a ≡ a ¯ . Moreover, we give a precise description of the decay rate and demonstrate that the system has the spectrum determined growth (SDG) property, i.e. the type of the induced semigroup coincides with the spectral bound for its generator.

Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8d3076ce08af6599cbfe3a55fa6bd4ad409c13e3

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In Computational Theory and Mathematics

In Control and Systems Engineering

In Mathematical Physics

Citations in the corpus

On the decay rates of Timoshenko system with second sound (2016, result): “For stabilization of Timoshenko systems via heat effect, Rivera and Racke [14] considered the following system8̂ :̂ 1'tt .”
Energy decay in a Timoshenko-type system of thermoelasticity of type III with different wave-propagation speeds (2013, background): “[13], Fernández Sare and Rivera [4], Messaoudi and Said-Houari [16], Rivera and Racke [18,19] and Mustafa and Messaoudi [20].”
Energy decay in a Timoshenko-type system of thermoelasticity of type III with different wave-propagation speeds (2008, background): “For Timenshinko systems in classical thermoelasticity, Rivera and Racke [13] considered ρ1ϕtt − σ(ϕx,ψ)x = 0 in (0,∞)× (0, L) ρ2ψtt − bψxx + k (ϕx + ψ) + γθx = 0 in (0,∞)× (0, L) ρ3θt − kθxx + γψtx = 0 in (0,∞)× (0, L) , (1.5) where ϕ,ψ, and θ are functions of (x, t) which model the transverse displ”
On the stabilization of the Timoshenko system by a weak nonlinear dissipation (2009, background): “In addition, Muñoz Rivera and Racke [13] treated a nonlinear system with damping effect through heat conduction of the form 1utt − (ux , x )x = 0 in (0, L)×(0,+∞) 2 t t −b xx +K (ux + )+ x = 0 in (0, L)×(0,+∞) 3 t −K xx + xt = 0 in (0, L)×(0,+∞) where is the difference temperature.”