The group property of a string system with time delay in boundary and the Riesz basis property of eigenvectors of the system are discussed in the present paper. It is proved that, when the feedback with delayτ > 0, the system also associates a C0 group, and its eigenvectors (generalized eigenvectors) form a Riesz basis in Hilbert spaceH. This result shows that time delay may destroy the stability of the system, but the group and Riesz basis properties are kept. As a consequence, the exact controllability of the system with boundary control is given.

In the present paper we consider the boundary feedback stabilization of a Timoshenko beam with both ends free. We propose boundary feedback control law that makes the closed loop system dissipative. Using asymptotic analysis techniques, we give explicit asymptotic formula of eigenvalues of the closed loop system, and prove the Riesz basis property of eigenvectors and generalized eigenvectors. By a detailed analysis of spectrum of the closed loop system, we show that the closed system is exponentially stable.

Suppose that {λn} is the set of zeros of a sine-type generating function of the exponentialsystem {eiλnt} in L2(0, T) and is separated. Levin and Golovin's classical theorem claims that {eiλnt} forms a Riesz basis for L2(0, T). In this article, we relate this result with Riesz basis generation of eigenvectors of the system operator of the linear time-invariant evolution equation in Hilbert spaces through its spectrum. A practically favorable necessary and sufficient condition for the separability of zeros of function of sine type is derived. The result is applied to get Riesz basis generation of a coupled string equation with joint dissipative feedback control.

We study the boundary stabilization of laminated beams with structural damping which describes the slip occurring at the interface of two-layered objects. By using an invertible matrix function with an eigenvalue parameter and an asymptotic technique for the first order matrix differential equation, we find out an explicit asymptotic formula for the matrix fundamental solutions and then carry out the asymptotic analyses for the eigenpairs. Furthermore, we prove that there is a sequence of generalized eigenfunctions that forms a Riesz basis in the state Hilbert space, and hence the spectrum determined growth condition holds. Furthermore, exponential stability of the closed-loop system can be deduced from the eigenvalue expressions. In particular, the semigroup generated by the system operator is a C0-group due to the fact that the three asymptotes of the spectrum are parallel to the imaginary axis.

This paper considers linear in5nite dimensional systems with C0-group generators and one-dimensional admissible input operators. The exact controllability and Riesz basis generation property are discussed. The corresponding results of Jacob and Zwart (Advances in Mathematical Systems Theory, Birkh:auser, Boston, MA, 2000) under the assumption of algebraic simplicity for eigenvalues of the generator are generalized to the case in which the eigenvalues are allowed to be algebraically multiple but with a uniform bound on the multiplicity.