Nonlinear damped Timoshenko systems with second sound—Global existence and exponential stability

Abstract

SUMMARYIn this paper, we consider nonlinear thermoelastic systems of Timoshenko type in a one-dimensional bounded domain. The system has two dissipative mechanisms being present in the equation for transverse displacement and rotation angle-a frictional damping and a dissipation through hyperbolic heat conduction modelled by Cattaneo's law, respectively. The global existence of small, smooth solutions and the exponential stability in linear and nonlinear cases are established.

Uniform decay in a Timoshenko-type system with past history

“…and proved an exponential decay similar to the one in [15] and [10]. For more results concerning well-posedness and controllability of Timoshenko systems, we refer the reader to Alabau-Boussouira [1], Fernández Sare and Racke [6], and Messaoudi and Mustafa [12].…”

“…Under appropriate conditions of σ , ρ i , b, k, γ , they proved several exponential decay results for the linearized system and a nonexponential stability result for the case of different wave speeds. Messaoudi et al [10] studied…”

Output-Based Stabilization of Timoshenko Beam with the Boundary Control and Input Distributed Delay

“…Besides the single beam, the network system of Timoshenko beam [11][12][13] and Timoshenko beam with structure memory [14] also were studied. Some nice results have been obtained for the closed-loop system, for example, Riesz basis property [10,12,13] and exponential stability [4,6,9,10,13]. We observe that there is a few work on the control delay problem.…”

“…Various different control strategy were used to stabilize the system, for instance, boundary control [4][5][6], partial internal control [7], pointwise control [8], and nonlinear control [9]. Besides the single beam, the network system of Timoshenko beam [11][12][13] and Timoshenko beam with structure memory [14] also were studied.…”

Exponential stability for a Timoshenko-type system with history

“…Also Messaoudi and Said-Houari [15] considered a Timoshenko-type system of thermoelasticity type III and proved an exponential decay similar to the one in [14,16]. Recently, Fernández Sare and Racke [5] considered hyperbolic Timoshenko-type vibrating systems that are coupled to a heat equation modeling an expectedly dissipative effect through heat conduction…”