On the stability of Timoshenko systems with Gurtin–Pipkin thermal law

We analyze the differential system{ρ1φtt−κ(φx+ψ)x=0ρ2ψtt−bψxx+κ(φx+ψ)+δθx=0ρ3θt−1β∫0∞g(s)θxx(t−s)ds+δψtx=0 describing a Timoshenko beam coupled with a temperature evolution of Gurtin–Pipkin type. A necessary and sufficient condition for exponential stability is established in terms of the structural parameters of the equations. In particular, we generalize previously known results on the Fourier–Timoshenko and the Cattaneo–Timoshenko beam models.