Boundary stabilization for a star-shaped network of variable coefficients strings linked by a point mass. (English) Zbl 1487.35068
Summary: This study is concerned with the pointwise stabilization for a star-shaped network of \(N\) variable coefficients strings connected at the common node by a point mass and subject to boundary feedback dampings at all extreme nodes. It is shown that the closed-loop system has a sequence of generalized eigenfunctions which forms a Riesz basis for the state Hilbert space. As a consequence, the spectrum-determined growth condition fulfills. In the meanwhile, the asymptotic expression of the spectrum is presented, and the exponential stability of the system is obtained by giving the optimal decay rate. We prove also that a phenomenon of lack of uniform stability occurs in the absence of damper at one extreme node. This paper reconfirmed the main stability results given by S. Hansen and E. Zuazua [SIAM J. Control Optim. 33, No. 5, 1357–1391 (1995; Zbl 0853.93018)] in a very particular case.
MSC:
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35L20 | Initial-boundary value problems for second-order hyperbolic equations |
| 35R02 | PDEs on graphs and networks (ramified or polygonal spaces) |
| 93D15 | Stabilization of systems by feedback |
| 93C20 | Control/observation systems governed by partial differential equations |
| 34B09 | Boundary eigenvalue problems for ordinary differential equations |
Keywords:
network of strings; Riesz basis; spectrum-determined growth condition; exponential stability; variable coefficients; spectral method; closed-loop systemCitations:
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