On the stabilization of Timoshenko systems with memory and different speeds of wave propagation

In this work we consider a one-dimensional Timoshenko system with different speeds of wave propagation and with only one control given by a viscoelastic term on the angular rotation equation. For a wide class of relaxation functions and for sufficiently regular initial data, we establish a general decay result for the energy of solution. Unlike the past history and internal feedback cases, the second energy is not necessarily decreasing. To overcome this difficulty, a precise estimate of the second energy, in terms of the initial data and the relaxation function, is proved.