Lack of Exponential Stability in Timoshenko Systems with Flat Memory Kernels

Lack of Exponential Stability in Timoshenko Systems with Flat Memory Kernels We analyze the decay properties of the solution semigroup generated by the linear Timoshenko system $$\begin{aligned} \left\{ \begin{array}{l} \rho _1 \varphi _{tt} -\kappa (\varphi _x +\psi )_x = 0\\ \displaystyle \rho _2 \psi _{tt} -b\psi _{xx} +\int _0^\infty \mu (s)\psi _{xx}(t-s)\,\mathrm{d}s+\kappa (\varphi _x +\psi )= 0 \end{array}\right. \end{aligned}$$ ρ 1 φ t t - κ ( φ x + ψ ) x = 0 ρ 2 ψ t t - b ψ x x + ∫ 0 ∞ μ ( s ) ψ x x ( t - s ) d s + κ ( φ x + ψ ) = 0 in presence of a flat (i.e. piecewise constant) memory kernel $$\mu $$ μ . In this situation, the uniform decay of the solutions does not occur, also when the two equations exhibit the same propagation speed. Applied Mathematics and Optimization Springer Journals

Lack of Exponential Stability in Timoshenko Systems with Flat Memory Kernels

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Springer US
Copyright © 2015 by Springer Science+Business Media New York
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics
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