Stability of Timoshenko systems with thermal coupling on the bending moment

Abstract

The Timoshenko system is a distinguished coupled pair of differential equations arising in mathematical elasticity. In the case of constant coefficients, if a damping is added in only one of its equations, it is well‐known that exponential stability holds if and only if the wave speeds of both equations are equal. In the present paper we study both non‐homogeneous and homogeneous thermoelastic problems where the model's coefficients are non‐constant and constants, respectively. Our main stability results are proved by means of a unified approach that combines local estimates of the resolvent equation in the semigroup framework with a recent control‐observability analysis for static systems. Therefore, our results complement all those on the linear case provided in [22], by extending the methodology employed in [4] to the case of Timoshenko systems with thermal coupling on the bending moment.

Exponential stability of Timoshenko-Gurtin-Pipkin systems with full thermal coupling

“…Instead, when the system is only partially damped (i.e. the effects of either θ or else ξ are neglected) exponential stability occurs only within the equal wave speed assumption ρ 1 b = ρ 2 k (see [1,6,22]).…”

Polynomial stability of piezoelectric beams with magnetic effect and tip body

“…With the help of Wely's theorem [37], we obtain that the growth bound of original system is zero as in the conservation system. Some research on this type of problem can be found in reference [1, 3, 5, 12, 17, 18, 20, 22–25, 27, 31, 32].…”

Laminated beams with thermoelasticity acting on the shear force

“…$$ Using the energy method, they proved the exponential decay of the system under the supposition of equality of the wave propagation speeds, given by $$$$ \chi := \frac{\kappa }{\rho_1}-\frac{b}{\rho_2}=0. $$$$ After the publication of Rivera and Racke's work, 2 many papers appeared in the literature 3–7 considering assumption A1 in studying coupled hyperbolic systems. However, leaving this paradigm, Almeida Júnior et al 8 proposed a new thermal coupling in the context of Timoshenko beam systems by considering assumption A2.…”

“…After the publication of Rivera and Racke's work, 2 many papers appeared in the literature [3][4][5][6][7] considering assumption A1 in studying coupled hyperbolic systems. However, leaving this paradigm, Almeida Júnior et al 8 proposed a new thermal coupling in the context of Timoshenko beam systems by considering assumption A2.…”