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On stability of hyperbolic thermoelastic Reissner-Mindlin-Timoshenko plates. (English) Zbl 1320.35059

Summary: In the present article, we consider a thermoelastic plate of Reissner-Mindlin-Timoshenko type with the hyperbolic heat conduction arising from Cattaneo’s law. In the absence of any additional mechanical dissipations, the system is often not even strongly stable unless restricted to the rotationally symmetric case, and so on. We present a well-posedness result for the linear problem under general mixed boundary conditions for the elastic and thermal parts. For the case of a clamped, thermally isolated plate, we show an exponential energy decay rate under a full damping for all elastic variables. Restricting the problem to the rotationally symmetric case, we further prove that a single frictional damping merely for the bending component is sufficient for exponential stability. To this end, we construct a Lyapunov functional incorporating the Bogovskii operator for irrotational vector fields, which we discuss in the appendix.

MSC:

35B35 Stability in context of PDEs
35L55 Higher-order hyperbolic systems
35Q74 PDEs in connection with mechanics of deformable solids
74D05 Linear constitutive equations for materials with memory
93D15 Stabilization of systems by feedback
93D20 Asymptotic stability in control theory
74K20 Plates

References:

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