Exponential Stability Of Systems Of Linear Timoshenko Type

In this paper, the stabilization problem of porous elastic solids in real world is considered. The kinetic behavior of porous solids is governed by equations of linear Timoshenko type. Under normal boundary conditions, due to the properties of porous solids, the system is generally asymptotically stable but not exponentially stable. So it should act boundary feedback controls in order to make the system to be exponentially stable. Here two systems are studied, one is the system with one end clamped and the other free, the other is both ends free system. Then properties such as well-posed-ness and stabilization of Timoshenko type systems with boundary controls are discussed.In fact, the system studied here is not normative Timoshenko system. Because it describes the behavior of porous solids, here exists restrictions to coefficients. The other difficult is that the formula of one dimension Timoshenko equation can't be used here. Then Birkhoff asymptotic evolution is adopted and the asymptotic values of eigenvalues of the closed system are gained. By means of spectrums of operators and Riesz basis, exponationl stability of the closed system is gained.Firstly, the system is normalized to an abstract Cauchy problem in Banach space. Then it shows that the operator determined by the system is dissipative and generates a C₀ semigroup, and hence the well-posed-ness of the system follows from the semigroup theory of bounded linear operators. Because the imaginary axis are in the resolvent set, the system is asymptotic stability. Secondly, using the solution of basic matrix and its asymptotic evolution, the asymptotic values of eigenvalues are got, which are isolated and lie in a strip area under certain condition. Moreover an auxiliary operator is introduced. Then by means of spectral properties of the auxiliary operator as well as its relationship with the system operator, it proves that there is a sequence of generalized eigenvector system of the studied system that forms a Riesz basis for Hilbert state space. Finally, the exponential stability of the closed loop system is obtained by use of the Riesz basis property and spectral distribution.Even though the purpose of this paper is to study a special system, this method can be generalized in the study of other models. It breaks a new path for the study of complicated systems.