Decay rates for delayed abstract thermoelastic systems with Cattaneo law

In this paper we consider a thermoelastic type system with Cattaneo’s law and internal time delay. Under a suitable assumption on the weight of delay, we prove that the exponential stability of this system is reduced to an observability estimate for the corresponding uncontrolled system. The proof of the main results uses the methodology introduced by Haraux (Port Math 46:245–258, 1989) and generalized by Ammari and Tucsnak (ESAIM COCV 6:361–386, 2001). Illustrating example is given.