Timoshenko systems with indefinite damping

Abstract

We consider the Timoshenko system in a bounded domain $(0,L)\subset{\Bbb R}^1$. The system has an indefinite damping mechanism, i.e. with a damping function $a=a(x)$ possibly changing sign, present 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 $\overline{a}=\int_0^La(x)\;dx>0$ and $\|a-\overline{a}\|_{L^2}