Exponential stability of a system of linear Timoshenko type with boundary controls
具有边界控制的线性Timoshenko型系统的指数稳定性

In this paper we consider the stabilization problem of porous elastic solids in real world. The kinetic behavior of porous solids is governed by equations of linear Timoshenko type which is usually asymptotically stable but not exponentially stable. We apply boundary velocity feedback controls to the system with both ends free, then examine the well-posedness and exponential stability of the closed loop system. Firstly, we obtain the well-posedness of the system by the semigroup theory of bounded linear operators. Secondly, we get the asymptotic values of eigenvalues of the system, which are isolated and lie in a strip area under certain condition. Moreover, we introduce an auxiliary operator, and then by means of its spectral properties and theory of perturbations of bounded linear operators to prove that there is a sequence of generalized eigenvector system which forms a Riesz basis for Hilbert state space. Finally, we obtain the exponential stability of the closed loop system by using the Riesz basis property and spectral distribution .