An Exponential Stability Result of a Timoshenko System with Thermoelasticity with Second Sound and in the Presence of Delay

In this paper, we consider a one-dimensional linear thermoelastic system of Timoshenko type with a delay, where the heat flux is given by Cattaneo's law. We prove an exponential decay result under a smallness condition on the delay and a stability number introduced first in Santos et al. (J Diff Eqs 253:2715-2733, 2012), using a method different from that of Santos et al. (J Diff Eqs 253:2715-2733, 2012). We also reproduce the polynomial decay of Santos et al. (J Diff Eqs 253:2715-2733, 2012) using the multiplier method in the case of absence of delay. The polynomial decay issue in the presence of a small delay is an open question.