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.

In this paper, we study the boundary stabilizing feedback control problem of Rayleigh beams that have non-homogeneous spatial parameters. We show that no matter how non-homogeneous the Rayleigh beam is, as long as it has positive mass density, sti9ness and mass moment of inertia, it can always be exponentially stabilized when the control parameters are properly chosen. The main steps are a detail asymptotic analysis of the spectrum of the system and the proving of that the generalized eigenfunctions of the feedback control system form a Riesz basis in the state Hilbert space. As a by-product, a conjecture in Guo (J. Optim. Theory Appl. 112(3) (2002) 529) is answered.

The exponential decay rate of a Timoshenko beam system with boundary damping is studied. By asymptotically analyzing the characteristic determinant of the system, we prove that the Timoshenko beam system is a Riesz system; hence, its decay rate is determined via its spectrum. As a consequence, by showing that the imaginary axis neither has an eigenvalue on it nor is an asymptote of the spectrum, we conclude that the system is exponentially stable.

This paper studies a linear hyperbolic system with static boundary condition that was first studied in Neves et al. [J. Funct. Anal. 67(1986) 320–344]. It is shown that the spectrum of the system consists of zeros of a sine-type function and the generalized eigenfunctions of the system constitute a Riesz basis with parentheses for the root subspace. The state space thereby decomposes into topological direct sum of root subspace and another invariant subspace in which the associated semigroup is superstable: that is to say, the semigroup is identical to zero after a finite time period.

Problems on the expansion of a semigroup and a criterion for being a Riesz basis are discussed in the present paper. Suppose that A is the generator of a C0 semigroup on a Hilbert space and σ(A) =σ 1(A) ∪ σ2(A) with σ2(A) is consisted of isolated eigenvalues distributed in a vertical strip. It is proved that if σ2(A) is separated and for each λ ∈σ 2(A), the dimension of its root subspace is uniformly bounded, then the generalized eigenvectors associated with 2(A) form an L-basis. Under different conditions on the Riesz projection, the expansion of a semigroup is studied. In particular, a simple criterion for the generalized eigenvectors forming a Riesz basis is given. As an application, a heat exchanger problem with boundary feedback is investigated. It is proved that the heat exchanger system is a Riesz system in a suitable state Hilbert space.

We study damped Euler–Bernoulli beams that have nonuniform thickness or density. These nonuniform features result in variable coefficient beam equations. We prove that despite the nonuniform features, the eigenfunctions of the beam form a Riesz basis and asymptotic behaviour of the beam system can be deduced without any restrictions on the sign of the damping. We also provide an answer to the frequently asked question on damping: 'how much more positive than negative should the damping be without disrupting the exponential stability?', and result in a criterion condition which ensures that the system is exponentially stable.