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

Abstract

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.

Boundary feedback stabilization and Riesz basis property of a 1-d first order hyperbolic linear system withL∞-coefficients

“…Secondly, it is shown that the closed loop system has a sequence of generalized eigenfunctions, which form a Riesz basis for the state space, and hence the spectrum-determined growth condition holds. This generalizes the results proved in [11,12] where the matrix K (respectively the diagonal matrix M) has continuously differentiable (respectively continuous) coefficients and also those in [16] and the application in [37] where the coefficients are constants. Now let us outline the content of this paper.…”

“…Furthermore, the assumption H.III ensures the dissipativity of the system operator. Recently, the Riesz basis property has been investigated for the system (1.1), with a diagonal matrix M and dynamic as well as static boundary conditions [11,12]. As pointed out, the smoothness of the coefficients K i and M i, j has been used in the works cited above.…”

“…) .By using Theorem 2.1, Corollary 2.1 and Theorem 2.6 of[12], it is seen that the operator A h 0 defined by (4.34) and (4.35) has the following properties:(i) the operator A h 0 is a discrete operator, in other words, for any λ h ∈ ρ(A h 0 ), the resolvent operator R(λ h , A h 0 ) is compact on (L 2 [0, 1]) 2 ;(ii) the operator A h 0 generates a C 0 -group in (L 2 [0, 1]) 2 ;(iii) there is a set of generalized eigenfunctions of A h 0 , which forms a Riesz basis with parentheses for…”

Spectral analysis and stability of thermoelastic Bresse system with second sound and boundary viscoelastic damping

“…From Theorems and , it is easy to obtain that the multiplicities of the eigenvalues of are uniformly bounded. Therefore, by , we obtain that the system associated with satisfies the spectrum‐determined‐growth condition, that is, , where is the growth order of and is the spectral bound of . The proof of Theorem is complete.…”

Singular functional differential equations of neutral type in Banach spaces

“…New results on nonautonomous neutral equations with atomic difference operators in Banach spaces can be found in [12,Section 6]. It is shown in [23,24] that neutral equation with atomic difference operators can be studied as hyperbolic systems (see also [10] for new results on such systems).…”