Ammari, Kais; Jellouli, Mohamed; Khenissi, Moez Stabilization of generic trees of strings. (English) Zbl 1064.93034 J. Dyn. Control Syst. 11, No. 2, 177-193 (2005). Summary: The authors study the energy decay of a tree-shaped network of vibrating elastic strings when the pointwise feedback acts in the root of the tree. They show that the strings are not exponentially stable in the energy space. Moreover, explicit polynomial decay estimates valid for regular initial data are given. Cited in 41 Documents MSC: 93D15 Stabilization of systems by feedback 35B37 PDE in connection with control problems (MSC2000) 93B07 Observability 74K05 Strings 35B40 Asymptotic behavior of solutions to PDEs 74M05 Control, switches and devices (“smart materials”) in solid mechanics 93C20 Control/observation systems governed by partial differential equations Keywords:tree-shaped network; explicit polynomial decay estimates PDFBibTeX XMLCite \textit{K. Ammari} et al., J. Dyn. Control Syst. 11, No. 2, 177--193 (2005; Zbl 1064.93034) Full Text: DOI References: [1] 1. K. Ammari and M. Tucsnak, Stabilization of second order evolution equations by a class of unbounded feedbacks. ESAIM, Control Optim. Calc. Var. 6 (2001), 361-386. · Zbl 0992.93039 · doi:10.1051/cocv:2001114 [2] 2. J. W. S. Cassels, An introduction to diophantine approximation. Cambridge Univ. Press, Cambridge (1966). · Zbl 0077.04801 [3] 3. R. D?ger, Observation and control of vibrations in tree-shaped networks of strings. SIAM J. Control 43 (2004), 590-623. · Zbl 1083.93022 · doi:10.1137/S0363012903421844 [4] 4. R. D?ger and E. Zuazua, Controllability of tree-shaped networks of vibrating strings. C. R. Acad. Sci. Paris. S?r. I Math. 332 (2001), 1087-1092. · Zbl 0990.93051 [5] 5. G. Doetsch, Introduction to the theory and application of the Laplace transformation. Springer-Verlag, Berlin (1974). · Zbl 0278.44001 [6] 6. J. Lagnese, G. Leugering, and E. J. P. G. Schmidt, Analysis and control of multi-link flexible structures. Systems and Control: Foundations and Applications. Birha?ser, Basel (1994). · Zbl 0810.73004 [7] 7. S. Lang, Introduction to diophantine approximations. Addison Wesley, New York (1966). · Zbl 0144.04005 [8] 8. G. Leugering and E. Zuazua, Exact controllability of generic trees. ESAIM Proc. 8 (2000), 95-105. · Zbl 0966.74049 · doi:10.1051/proc:2000007 [9] 9. J. L. Lions and E. Magenes, Probl?mes aux limites non homog?nes et applications. Dunod, Paris (1968). [10] 10. S. Ni?aise, Spectre des r?seaux topologiques finis. Bull. Sci. Math. 111 (1987), 401-413. [11] 11. A. Pazy, Semigroups of linear operators and applications to partial differential equations. Springer-Verlag, New York (1983). · Zbl 0516.47023 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.