In this paper we study the Bresse system with frictional dissipation working only on the angle displacement. Our main result is to prove that this dissipative mechanism is enough to stabilize exponentially the whole system provided the velocities of waves propagations are the same. This result is significative only from the mathematical point of view since in practice the velocities of waves propagations are always different. In that direction we show that when the velocities are not the same, the system is not exponentially stable and we prove that the solution in this case goes to zero polynomially, with rates that can be improved by taking more regular initial data. Finally, we give some numerical result to verify our analytical results.

In this work, we consider the Timoshenko beam model with second sound. We introduce a new number χ 0 that characterizes the exponential decay. We prove that the corresponding semigroup associated to the system is exponentially stable if and only if χ 0 = 0. Otherwise there is a lack of exponential stability. In this case we prove that the semigroup decays as t −1/2 . Moreover we show that the rate is optimal.

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In this paper, we consider the Timoshenko systems with frictional dissipation working only on the vertical displacement. We prove that the system is exponentially stable if and only if the wave speeds are the same. On the contrary, we show that the Timoshenko systems is polynomially stable giving the optimal decay rate. Copyright © 2013 John Wiley & Sons, Ltd.

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