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 the present paper the stabilization problem of porous elastic solids is considered. The kinetic behavior of porous solids is governed by equations of linear Timoshenko type which is generally asymptotically stable but not exponentially stable. For the exponential stability, boundary velocity feedback controls are applied with one end clamped and the other free. Firstly, it is shown that the operator determined by the system is dissipative and generates a $C_0$ semigroup. Hence the well-posed-ness of the system follows from the semigroup theory of bounded linear operators. Secondly, the asymptotic behavior of eigenvalues of $\mathcal{A}$ is obtained under certain condition.Moreover by using an auxiliary operator $\mathcal{A}$$_0$, and by means of spectral properties of $\mathcal{A}$$_0$ , it is proven that there is a sequence of generalized eigenvectors of $\mathcal{A}$ which forms a Riesz basis for Hilbert state space. Finally, the exponential stability of the closed loop system is givenby use of the Riesz basis property and spectral distribution of $\mathcal{A}$.