Bresse system with indefinite damping

We consider the Bresse system in a bounded domain ( 0 , L ) ⊂ R 1 . The system has an indefinite damping mechanism, i.e. with a damping function a = a ( x ) possibly changing sign, presented only in the equation for the rotation angle. We shall prove that the system is still exponentially stable under the same conditions as in the positive constant damping case, and provided a ¯ = 1 L ∫ 0 L a ( x ) d x > 0 and ‖ a − a ¯ ‖ L 2 < ϵ , for ϵ small enough. In the arguments, we shall also give a new proof of exponential stability for the constant case a = a ¯ .