Riesz basis generation and boundary stabilization of two strings connected by a point mass with variable coefficients

Abstract

In this paper, we study the Riesz basis property and the problem of stabilization of two vibrating strings connected by a point mass with variable physical coefficients under a boundary feedback control acts at one extreme point and Dirichlet boundary condition on the other end. It is shown that the system has a sequence of generalized eigenfunctions which forms a Riesz basis for the state Hilbert space. By a detailed spectral analysis, it is proved that this hybrid system is asymptotically stable but not exponentially stable.

Boundary stabilization for a star-shaped network of variable coefficients strings linked by a point mass

“…If K N = 0, the proof of the result on the localization of the spectrum is similar to that of the first case, and the evidence when N = 2 is that of [16,Prop. 2].…”

“…Sometimes, the use of Shkalikov's theory [42] is helpful when a spectral parameter appears in the boundary conditions of the associated spectral problem (see e.g., [12,23]). In the majority of cases, the application of the abstract result due to Xu and Yung [45] be the perfect solution to get the Riesz basis generation for string equations [16], for thermo-elastic systems [30], or for Timoshenko beam systems [31]. For more details on Riesz bases, see Akhiezer and Glazman [1] and Gohberg and Krein [22].…”

“…Due to the paper of Lee and You [33] about the stability of string with a tip mass at a part of the boundary, the attention of many researchers has been caught towards this problem (for instance, see [16,32,34,36] and the references therein). In particular, If N = 2, ρ j , σ j are constants and q j ≡ 0 (j = 1, 2) in System (1), we come back to the case of the famous paper of Hansen and Zuazua [32,Sec.…”

“…An exponential stability result was given with an appropriate choice of damping constants. Recently, Ben Amara and Boughamda [16] considered two non-homogeneous strings connected by a point mass and subject to a boundary feedback control acting at one extreme point (i.e., N = 2 and K 2 = 0 in System (1)). In that paper, the authors showed that there exists a sequence of eigenvectors which forms a Riesz basis for the state space.…”