Abstract
We consider a tree-shaped network of vibrating elastic strings, with feedback acting on the root of the tree. Using the d’Alembert representation formula, we show that the input-output map is bounded, i.e. this system is a well-posed system in the sense of G. Weiss (Trans. Am. Math. Soc. 342 (1994), 827–854). As a consequence we prove that the strings networks are not exponentially stable in the energy space. Moreover, we give explicit polynomial decay estimates valid for regular initial data.
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References
K. Ammari, M. Jellouli: Stabilization of star-shaped networks of strings. Differ. Integral Equations 17 (2004), 1395–1410.
K. Ammari, M. Jellouli, and M. Khenissi: Stabilization of generic trees of strings. J. Dyn. Control Syst. 11 (2005), 177–193.
K. Ammari, M. Tucsnak: Stabilization of second order evolution equations by a class of unbounded feedbacks. ESAIM, Control Optim. Calc. Var. 6 (2001), 361–386.
J. von Below: Classical solvability of linear parabolic equations in networks. J. Differ. Equations 52 (1988), 316–337.
R. Dáger: Observation and control of vibrations in tree-shaped networks of strings. SIAM. J. Control Optim. 43 (2004), 590–623.
R. Dáger, E. Zuazua: Wave propagation, observation and control in 1-d flexible multi-structures. Mathématiques et Applications, Vol. 50. Springer-Verlag, Berlin, 2006.
R. Dáger, E. Zuazua: Controllability of star-shaped networks of strings. C. R. Acad. Sci. Paris 332 (2001), 621–626.
R. Dáger, E. Zuazua: Controllability of tree-shaped networks of vibrating strings. C. R. Acad. Sci. Paris 332 (2001), 1087–1092.
J. Lagnese, G. Leugering, and E. J. P. G. Schmidt: Modeling, Analysis of Dynamic Elastic Multi-link Structures. Birkhäuser-Verlag, Boston-Basel-Berlin, 1994.
I. Lasiecka, J.-L. Lions, and R. Triggiani: Nonhomogeneous boundary value problems for second-order hyperbolic generators. J. Math. Pures Appl. 65 (1986), 92–149.
J.-L. Lions, E. Magenes: Problèmes aux limites non homogènes et applications. Dunod, Paris, 1968.
J. L. Lions: Contrôle des systèmes distribués singuliers. Gauthier-Villars, Paris, 1983.
A. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York, 1983.
E. J. P. G. Schmidt: On the modelling and exact controllability of networks of vibrating strings. SIAM J. Control Optim. 30 (1992), 229–245.
G. Weiss: Transfer functions of regular linear systems. Part I. Characterizations of regularity. Trans. Am. Math. Soc. 342 (1994), 827–854.
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Ammari, K., Jellouli, M. Remark on stabilization of tree-shaped networks of strings. Appl Math 52, 327–343 (2007). https://doi.org/10.1007/s10492-007-0018-1
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DOI: https://doi.org/10.1007/s10492-007-0018-1