A Sufficient Condition on Riesz Basis with Parentheses of Non--Self-Adjoint Operator and Application to a Serially Connected String System under Joint Feedbacks

Expansion of solution in terms of generalized eigenfunctions for a hyperbolic system with static boundary condition

“…In control theory, the Riesz basis is studied in the context of stabilization of linear infinite-dimensional systemẋ(t) = Ax(t) in some Hilbert space H, where A is the generator of a C 0 -semigroup on H. The system is called a Riesz spectral system [4] if there is a set of (generalized) eigenvectors of A, which forms a Riesz basis for H. For a Riesz spectral system, not only the stability of the system is determined by the spectrum of A, which is referred to as the spectrum-determined growth condition, but also the dynamic behavior of the system can be described by eigenpairs under expansion of nonharmonic Fourier series. Some examples can be found in [6][7][8][9]11,21].…”

Eliciting harmonics on strings

“…It is now a simple matter to confirm that equality holds in (4.6). We note that an alternate route to completeness has recently been established in the works of Guo and Xie [14] and Xu and Guo [33].…”

Riesz Basis Property and Stability of Planar Networks of Controlled Strings

“…Similarly, |G(λ)| is bounded on the sector with boundary rays λ = α, λ ≤ 0 and λ = 0, λ ≤ α by Phragmén-Lindelöf Theorem. Therefore, it yields that |G(λ)| is bounded on C, i.e., To study the Riesz basis generation of the (generalized) eigenvectors of A, we give some definitions about Riesz basis, Riesz basis with parentheses and Riesz basis of subspaces (see [20][21][22][23]). …”

“…Besides the methods mentioned above, some researchers used the frequency analysis method to study the control problems, for example, [16][17][18] for serially connected strings and Timoshenko beams, [15] for symmetric tree-shaped Euler-Bernoulli beam networks. We observe that for a controlled network, the operator determined by the closed loop system is non-normal, whose spectral analysis and basis generation of the root vectors are tough problems in operator theory.…”