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On the stability of a star-shaped network of variable coefficients strings under joint damping. (English) Zbl 1486.35045

Summary: This paper is devoted to the stabilization problem of a star-shaped network of strings with variable physical coefficients, under a feedback control at the common node and an extreme node, and subject to Dirichlet or Neumann boundary conditions on the two other nodes. We show that the system has a sequence of (generalized) eigenfunctions, which forms a Riesz basis with parentheses for the state Hilbert space. The spectrum-determined growth condition fulfills. We prove the exponential stability of the system under some conditions on the physical coefficients. Our approach is based on the spectrum mapping theorem, and our method not only leads to an improved result, it is also simpler than the method used by Y. N. Guo and G. Q. Xu [Glasg. Math. J. 53, No. 3, 481–499 (2011; Zbl 1227.93096)] in easier situation. At the end, we prove that a phenomenon of lack of uniform stability occurs when a point mass is concentrated in the common node.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35L53 Initial-boundary value problems for second-order hyperbolic systems
47B06 Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
93C15 Control/observation systems governed by ordinary differential equations
93C20 Control/observation systems governed by partial differential equations

Citations:

Zbl 1227.93096
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References:

[1] Ammari, K.; Jellouli, M.; Khenissi, M., Stabilization of generic trees of strings, J. Dyn. Contin. Syst., 11, 177-193 (2005) · Zbl 1064.93034 · doi:10.1007/s10883-005-4169-7
[2] Ammari, K.; Jellouli, M., Remark in stabilization of tree-shaped networks of strings, Appl. Math., 52, 4, 327-343 (2007) · Zbl 1164.93315 · doi:10.1007/s10492-007-0018-1
[3] Ammari, K.; Jellouli, M., Stabilization of star-shaped networks of strings, Differ. Integr. Equ., 17, 1395-1410 (2004) · Zbl 1150.93537
[4] Ammari, K., Henrot, A., Tucsnak, M.: Asymptotic behaviour of the solutions and optimal location of the actuator for the pointwise stabilization of a string. Asympt. Anal. 28(3, 4), 215-240 (2001) · Zbl 0994.35030
[5] Ammari, K.; Liu, Z.; Shel, F., Stability of the wave equations on a tree with local Kelvin-Voigt damping, Semigroup Forum, 100, 364-382 (2020) · Zbl 1433.35388 · doi:10.1007/s00233-019-10064-7
[6] Ammari, K.; Mercier, D., Boundary feedback stabilization of a chain of serially connected strings, Evol. Equ. Control Theory, 1, 1-19 (2015) · Zbl 1433.93104
[7] Ammari, K.; Shel, F.; Vanninathan, M., Feedback stabilization of a simplified model of fluid-structure interaction on a tree, Asymptot. Anal., 103, 33-55 (2017) · Zbl 1378.35293
[8] Assel, R.; Jellouli, M.; Khenissi, M., Optimal decay rate for the local energy of a unbounded network, J. Differ. Equ., 261, 7, 4030-4054 (2016) · Zbl 1348.35286 · doi:10.1016/j.jde.2016.06.016
[9] Avdonin, S.; Edward, J., Exact controllability for string with attached masses, SIAM J. Control. Optim., 56, 945-980 (2018) · Zbl 1390.93395 · doi:10.1137/15M1029333
[10] Avdonin, S., Ivanov, S.: Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems. Cambridge University Press, Cambridge (1995) · Zbl 0866.93001
[11] Ben Amara, J.; Beldi, E., Boundary controllability of two vibrating strings connected by a point mass with variable coefficients, SIAM J. Control. Optim., 57, 5, 3360-3387 (2019) · Zbl 1423.35363 · doi:10.1137/16M1100496
[12] Ben Amara, J.; Beldi, E., Simultaneous controllability of two vibrating strings with variable coefficients, Evol. Equ. Control Theory, 8, 4, 687-694 (2019) · Zbl 1425.93035 · doi:10.3934/eect.2019032
[13] Ben Amara, J.; Boughamda, W., Exponential stability of two strings under joint damping with variable coefficients, Syst. Cont. Lett., 141 (2020) · Zbl 1447.93297 · doi:10.1016/j.sysconle.2020.104709
[14] Ben Amara, J.; Boughamda, W., Riesz basis generation and boundary stabilization of two strings connected by a point mass with variable coefficients, Math. Meth. Appl. Sci., 43, 5, 2322-2336 (2020) · Zbl 1447.35049 · doi:10.1002/mma.6043
[15] Chen, G.; Coleman, M.; West, HH, Pointwise stabilization in the middle of the span for second order systems, nonuniform and uniform exponential decay of solutions, SIAM J. Appl. Math., 47, 4, 751-780 (1987) · Zbl 0641.93047 · doi:10.1137/0147052
[16] Chen, S.; Liu, K.; Liu, Z., Spectrum and stability for elastic systems with global or local Kelvin-Voigt damping, SIAM J. Appl. Math., 59, 651-668 (1999) · Zbl 0940.34036 · doi:10.1137/S0036139996308054
[17] Conway, J., Functions of One Complex Variable (1978), New York, Berlin: Springer, New York, Berlin · doi:10.1007/978-1-4612-6313-5
[18] Fedoryuk, M.V.: Asymptotic Analysis: Linear Ordinary Differential Equations. Springer, Berlin (1993) · Zbl 0782.34001
[19] Liu, K.; Liu, Z.; Zhang, Q., Eventual differentiability of a string with local Kelvin-Voigt damping SIAM, J. Control Optim., 54, 1859-1871 (2016) · Zbl 1343.35025 · doi:10.1137/15M1049385
[20] Liu, Z.; Rao, B., Frequency domain characterization of rational decay rate for solution of linear evolution equations, Z. Angew. Math. Phys., 56, 630-644 (2005) · Zbl 1100.47036 · doi:10.1007/s00033-004-3073-4
[21] Gohberg, I.C., Krein, M.G.: Introduction to the Theory of Linear Nonselfadjoint Operators. Trans. Math. Monogr. 18, AMS, Providence, RI (1969) · Zbl 0181.13503
[22] Gomilko, A.; Pivovarchik, V., On basis properties of a part of eigenfunctions of the problem of vibrations of a smooth inhomogeneous string damped at the midpoint, Math. Nachr., 245, 1, 72-93 (2002) · Zbl 1023.34023 · doi:10.1002/1522-2616(200211)245:1<72::AID-MANA72>3.0.CO;2-X
[23] Guo, B.Z., Wang, J.M.: Control of Wave and Beam PDEs: The Riesz Basis Approach, 596. Springer, Cham (2019) · Zbl 1426.35002
[24] Guo, BZ; Xu, GQ, Expansion of solution in terms of generalized eigenfunctions for a hyperbolic system with static boundary condition, J. Funct. Anal., 231, 245-268 (2006) · Zbl 1153.35368 · doi:10.1016/j.jfa.2005.02.006
[25] Guo, BZ; Zhu, WD, On the energy decay of two coupled strings through a joint damper, J. Sound Vib., 203, 3, 447-455 (1997) · doi:10.1006/jsvi.1996.0853
[26] Guo, YN; Xu, GQ, Exponential stabilisation of a tree-shaped network of strings with variable coefficients, Glasgow Math. J., 53, 3, 481-499 (2011) · Zbl 1227.93096 · doi:10.1017/S0017089511000085
[27] Levitan, B.M., Sargsjan, I.C.: Introduction to Spectral Theory: Selfadjoint Ordinary Differential Operators. AMS (1975) · Zbl 0302.47036
[28] Liu, KS, Energy decay problems in the design of a point stabilizer for coupled string vibrating systems, SIAM J. Control. Optim., 26, 1348-1356 (1988) · Zbl 0662.93054 · doi:10.1137/0326076
[29] Liu, KS; Huang, FL; Chen, G., Exponential stability analysis of a long chain of coupled vibrating strings with dissipative linkage, SIAM J. Appl. Math., 49, 6, 1694-1707 (1989) · Zbl 0685.93054 · doi:10.1137/0149102
[30] Lyubich, YI; Phóng, VQ, Asymptotic stability of linear differential equations in Banach spaces, Stud. Math., 88, 37-42 (1988) · Zbl 0639.34050 · doi:10.4064/sm-88-1-37-42
[31] Markus, A.S.: Introduction to the Spectral Theory of Polynomial Operator Pencils. AMS Transl. Math. (1988) · Zbl 0678.47005
[32] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations (1983), Berlin: Springer, Berlin · Zbl 0516.47023 · doi:10.1007/978-1-4612-5561-1
[33] Rzepnicki, Ł.; Schnaubelt, R., Polynomial stability for a system of coupled strings, Bull. Lond. Math. Soc., 50, 6, 1117-1136 (2018) · Zbl 1406.35185 · doi:10.1112/blms.12212
[34] Xu, GQ; Han, ZJ; Yung, SP, Riesz basis property of serially connected Timoshenko beams, Int. J. Control, 80, 470-485 (2007) · Zbl 1120.93026 · doi:10.1080/00207170601100904
[35] Xu, GQ; Yung, S., The expansion of semigroup and a Riesz basis criterion, J. Differ. Equ., 210, 1-24 (2005) · Zbl 1131.47042 · doi:10.1016/j.jde.2004.09.015
[36] Young, R.M.: An Introduction to Nonharmonic Fourier Series. Academic Press, London (1980) · Zbl 0493.42001
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