In this paper we study the dynamic feedback stability for some simplified model of fluid–structure interaction on a tree. We prove that, under some conditions, the energy of the solutions of the dissipative system decay exponentially to zero when the time tends to infinity. Our technique is based on a frequency domain method and a special analysis for the resolvent.
A.B.Abdallah and F.Shel, Exponential stability of a general network of 1-d thermoelastic materials, Mathematical Control and Related Fields2 (2012), 1–16. doi:10.3934/mcrf.2012.2.1.
2.
K.Ammari, A.Henrot and M.Tucsnak, Asymptotic behaviour of the solutions and optimal location of the actuator for the pointwise stabilization of a string, Asymptotic Analysis28 (2001), 215–240.
3.
K.Ammari and M.Jellouli, Stabilization of star-shaped networks of strings, Differential and Integral Equations17 (2004), 1395–1410.
4.
K.Ammari, M.Jellouli and M.Khenissi, Stabilization of generic trees of strings, J. Dynamical Control Systems11 (2005), 177–193. doi:10.1007/s10883-005-4169-7.
5.
K.Ammari and S.Nicaise, Stabilization of Elastic Systems by Collocated Feedback, Lecture Notes in Mathematics, Vol. 2124Springer, Cham, 2015.
6.
C.Batty and Y.Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann.347 (2010), 455–478. doi:10.1007/s00208-009-0439-0.
7.
H.Brezis, Analyse Fonctionnelle, Théorie et Applications, Masson, Paris, 1983.
8.
C.Conca, J.Planchard and M.Vanninathan, Fluids and Periodic Structures, Research in Applied Mathematics, Vol. 38, John Wiley & Sons, Masson, 1995.
9.
R.Dàger and E.Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-Structures, Mathématiques & Applications, Springer-Verlag, Berlin, 2006.
10.
S.Ervedoza and M.Vanninathan, Controllability of a simplified model of fluid–structure interaction, ESAIM Control Optim. Calc. Var.20 (2014), 547–575. doi:10.1051/cocv/2013075.
11.
F.L.Huang, Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Diff. Eqs.1 (1985), 43–56.
12.
Y.Liu, T.Takahashi and M.Tucsnak, Single input of a simplified fluid–structure interaction model, ESAIM Control Optim. Calc. Var.19 (2013), 20–42. doi:10.1051/cocv/2011196.
13.
D.Mercier and V.Régnier, Control of a network of Euler–Bernoulli beams, J. Math. Anal. Appl.342 (2008), 874–894. doi:10.1016/j.jmaa.2007.12.062.
14.
A.Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations, Springer, 1983.
15.
J.Prüss, On the spectrum of -semigroups, Trans. Amer. Math. Soc.284 (1984), 847–857. doi:10.2307/1999112.
F.Shel, Exponential stability of a network of elastic and thermoelastic materials, Math. Meth. in Appl. Sci.36 (2013), 869–879. doi:10.1002/mma.2644.
18.
M.Tucsnak and M.Vanninathan, Locally distributed control for a model of fluid–structure interaction, System and Control Letters58 (2009), 547–552. doi:10.1016/j.sysconle.2009.03.002.
19.
J.L.Vazquez and E.Zuazua, Large time behavior for a simplified 1D model of fluid–solid interaction, Comm. Partial Differential Equations28 (2003), 1705–1738. doi:10.1081/PDE-120024530.
20.
J.L.Vazquez and E.Zuazua, Lack of collision in a simplified 1D model for fluid–solid interaction, Math. Models Methods Appl. Sci.16 (2006), 637–678. doi:10.1142/S0218202506001303.
21.
S.M.Venuti, Modeling, analysis and computation of fluid structure interaction models for biological systems, SIAM Undergraduate Research Online3 (2010), 1–17. doi:10.1137/09S010496.
