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June  2011, 6(2): 297-327. doi: 10.3934/nhm.2011.6.297

Dynamical behavior of networks of non-uniform Timoshenko beams system with boundary time-delay inputs

1. 

Department of Mathematics, Tianjin University, Tianjin 300072

Received  June 2010 Revised  May 2011 Published  May 2011

The dynamical stability of planar networks of non-uniform Timoshenko beams system is considered. Suppose that the displacement and rotational angle is continuous at the common vertex of this network and the bending moment and shear force satisfies Kirchhoff's laws, respectively. Time-delay terms exist in control inputs at exterior vertices. The feedback control laws are designed to stabilize this kind of networks system. Then it is proved that the corresponding closed loop system is well-posed. Under certain conditions, the asymptotic stability of this system is shown. By a complete spectral analysis, the spectrum-determined-growth condition is proved to be satisfied for this system. Finally, the exponential stability of this system is discussed for a special case and some simulations are given to support these results.
Citation: Zhong-Jie Han, Gen-Qi Xu. Dynamical behavior of networks of non-uniform Timoshenko beams system with boundary time-delay inputs. Networks & Heterogeneous Media, 2011, 6 (2) : 297-327. doi: 10.3934/nhm.2011.6.297
References:
[1]

R. A. Adams, "Sobolev Spaces,", Pure and Applied Mathematics, 65 (1975). Google Scholar

[2]

K. Ammari, Asymptotic behaviour of some elastic planar networks of Bernoulli-Euler beams,, Appl. Anal., 86 (2007), 1529. doi: 10.1080/00036810701734113. Google Scholar

[3]

K. Ammari and M. Jellouli, Stabilization of star-shaped networks of strings,, Differential and Integral Equations, 17 (2004), 1395. Google Scholar

[4]

K. Ammari and M. Jellouli, Remark on stabilization of tree-shaped networks of strings,, Applications of Mathematics, 52 (2007), 327. doi: 10.1007/s10492-007-0018-1. Google Scholar

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K. Ammari, M. Jellouli and M. Khenissi, Stabilization of generic trees of strings,, Journal of Dynamical and Control Systems, 11 (2005), 177. doi: 10.1007/s10883-005-4169-7. Google Scholar

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S. A. Avdonin and S. A. Ivanov, "Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems,", Cambridge University Press, (1995). Google Scholar

[7]

J. W. Brown and R. V. Churchill, "Complex Variables and Applications," Seventh Edition,, China Machine Press, (2004). Google Scholar

[8]

P. G. Casazza and G. Kutyniok, Frames of subspaces,, Contemp. Math., 345 (2004), 87. Google Scholar

[9]

G. Chen, M. Coleman and H. H. West, Pointwise stabilization in the middle of the span for second order systems, nonuniform and uniform exponential decay of solutions,, SIAM J. Appl. Math., 47 (1987), 751. doi: 10.1137/0147052. Google Scholar

[10]

G. Chen, M. Delfour, A. Krall and G. Payre, Modeling, stabilization and control of seraially connected beams,, SIAM J. Control Optim, 25 (1987), 526. doi: 10.1137/0325029. Google Scholar

[11]

G. Chen, S. G. Krantz, D. L. Russell, C. E. Wayne, H. H. West and M. P. Coleman, Analysis, designs, and behavior of dissipative joints for coupled beams,, SIAM J. Appl. Math., 49 (1989), 1665. doi: 10.1137/0149101. Google Scholar

[12]

R. Datko, Two examples of ill-posedness with respect to small time delays in stabilized elastic systems,, IEEE Trans. Automatic Control, 38 (1993), 163. doi: 10.1109/9.186332. Google Scholar

[13]

R. Datko, J. Lagnese and M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations,, SIAM J. Control Optim., 24 (1986), 152. doi: 10.1137/0324007. Google Scholar

[14]

I. C. Gohberg and M. G. Krein, "Introduction to the Theory of Linear Nonselfadjoint Operators,", AMS Transl. Math. Monographs, (1969). Google Scholar

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B. Z. Guo and Y. Xie, A sufficient condition on Riesz basis with parentheses of non-self-adjoint operator and application to a serially connected string system under joint feedbacks,, SIAM J. Control Optim., 43 (2004), 1234. doi: 10.1137/S0363012902420352. Google Scholar

[16]

B. Z. Guo and K. Y. Yang, Output feedback stabilization of a one-dimensional Schrödinger equation by boundary observation with time delay,, IEEE Transactions on Automatic Control, 55 (2010), 1226. doi: 10.1109/TAC.2010.2051070. Google Scholar

[17]

Z. J. Han and L. Wang, Riesz basis property and stability of planar networks of controlled strings,, Acta Appl. Math., 110 (2010), 511. doi: 10.1007/s10440-009-9459-8. Google Scholar

[18]

Z. J. Han and G. Q. Xu, Spectrum and dynamical behavior of a kind of planar network of non-uniform strings with non-collocated feedbacks,, Networks and Heterogeneous Media, 5 (2010), 315. doi: 10.3934/nhm.2010.5.315. Google Scholar

[19]

Z. J. Han and G. Q. Xu, Exponential stabilisation of a simple tree-shaped network of Timoshenko beams system,, International Journal of Control, 83 (2010), 1485. doi: 10.1080/00207179.2010.481767. Google Scholar

[20]

Z. J. Han, G. Q. Xu, Stabilization and Riesz basis of a star-shaped network of Timoshenko beams,, Journal of Dynamical and Control Systems, 16 (2010), 227. doi: 10.1007/s10883-010-9091-y. Google Scholar

[21]

Z. J. Han and G. Q. Xu, Stabilization and Riesz basis property of two serially connected Timoshenko beams system,, Z. Angew. Math. Mech., 89 (2009), 962. doi: 10.1002/zamm.200800176. Google Scholar

[22]

Z. J. Han and G. Q. Xu, Exponential stability of Timoshenko beam system with delay terms in boundary feedbacks,, ESAIM: Control, 17 (2011), 552. doi: 10.1051/cocv/2010009. Google Scholar

[23]

J. Lagnese, G. Leugering and E. J. P. G. Schmidt, "Modeling, Analysis of Dynamic Elastic Multi-Link Structures,", Birkhäuser-Verlag, (1994). Google Scholar

[24]

J. S. Liang and Y. Q. Chen, Boundary control of wave equations with delayed boundary measurement,, Proceedings of IEEE International Conference on Robotics and Biomimetics, (2004), 849. doi: 10.1109/ROBIO.2004.1521895. Google Scholar

[25]

J. S. Liang, Y. Q. Chen and B. Z. Guo, A new boundary control method for beam equation with delayed boundary measurement using modified smith predictors,, Proceedings of the 42nd IEEE Conference on Decision and Control, (2003), 809. Google Scholar

[26]

Yu. I. Lyubich and V. Q. Phóng, Asymptotic stability of linear differential equations in Banach spaces,, Studia Math., 88 (1988), 34. Google Scholar

[27]

R. Mennicken and M. Möller, "Non-self-adjoint Boundary Eigenvalue Problem,", North-Holland Mathematics Studies, 192 (2003). Google Scholar

[28]

D. Mercier, Spectrum analysis of a serially connected Euler-Bernoulli beams problems,, Networks and Heterogeneous Media, 4 (2009), 709. doi: 10.3934/nhm.2009.4.709. Google Scholar

[29]

D. Mercier and V. Régnier, Spectrum of a network of Euler-Bernoulli beams,, Journal of Mathematical Analysis and Applications, 337 (2008), 174. doi: 10.1016/j.jmaa.2007.03.080. Google Scholar

[30]

D. Mercier and V. Régnier, Control of a network of Euler-Bernoulli beams,, Journal of Mathematical Analysis and Applications, 342 (2008), 874. doi: 10.1016/j.jmaa.2007.12.062. Google Scholar

[31]

W. Michiels and S. I. Niculescu, "Stability and Stabilization of Time-Delay Systems. An Eigenvalue-Based Approach,", Society for Industrial and Applied Mathematics, (2007). doi: 10.1137/1.9780898718645. Google Scholar

[32]

O. Morgul, On the stabilization and stability robustness against small delays of some damped wave equation,, IEEE Trans. Automatic Control, 40 (1995), 1626. doi: 10.1109/9.412634. Google Scholar

[33]

S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks,, SIAM J. Control Optim., 45 (2006), 1561. doi: 10.1137/060648891. Google Scholar

[34]

S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay,, Differential and Integral Equations, 21 (2008), 935. Google Scholar

[35]

S. Nicaise and J. Valein, Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks,, Networks and Heterogeneous Media, 2 (2007), 425. doi: 10.3934/nhm.2007.2.425. Google Scholar

[36]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Springer-Verlag, (1983). Google Scholar

[37]

A. A. Shkalikov, Boundary problems for ordinary differential equations with parameter in the boundary conditions,, J. Soviet Math., 33 (1986), 1311. doi: 10.1007/BF01084754. Google Scholar

[38]

K. Sriram and M. S. Gopinathan, A two variable delay model for the circadian rhythm of Neurospora crassa,, J. Theor. Biol., 231 (2004), 23. doi: 10.1016/j.jtbi.2004.04.006. Google Scholar

[39]

J. Srividhya and M. S. Gopinathan, A simple time delay model for eukaryotic cell cycle,, Journal of Theoretical Biology, 241 (2006), 617. doi: 10.1016/j.jtbi.2005.12.020. Google Scholar

[40]

H. Suh and Z. Bien, Use of time-delay actions in the controller design,, IEEE Trans. Automatic Control, 25 (1980), 600. doi: 10.1109/TAC.1980.1102347. Google Scholar

[41]

S. Timoshenko, "Vibration Problems in Engineering,", Van Norstrand, (1955). Google Scholar

[42]

J. Valein and E. Zuazua, Stabilization of the wave equation on 1-d networks,, SIAM J. Contr. Optim, 48 (2009), 2771. doi: 10.1137/080733590. Google Scholar

[43]

Q. P. Vu, J. M. Wang, G. Q. Xu and S. P. Yung, Spectral analysis and system of fundamental solutions for Timoshenko beams,, Appl. Math. Lett., 18 (2005), 127. doi: 10.1016/j.aml.2004.09.001. Google Scholar

[44]

J. M. Wang and B. Z. Guo, Riesz basis and stabilization for the flexible structure of a symmetric tree-shaped beam network,, Math. Meth. Appl. Sci., 31 (2008), 289. doi: 10.1002/mma.909. Google Scholar

[45]

G. Q. Xu, B. Z. Guo, Riesz basis property of evolution equations in Hilbert spaces and application to a coupled string equation,, SIAM J. Control Optim., 42 (2003), 966. doi: 10.1137/S0363012901400081. Google Scholar

[46]

G. Q. Xu, Z. J. Han and S. P. Yung, Riesz basis property of serially connected Timoshenko beams,, International Journal of Control, 80 (2007), 470. doi: 10.1080/00207170601100904. Google Scholar

[47]

G. Q. Xu and J. G. Jia, The group and Riesz basis properties of string systems with time delay and exact controllability with boundary control,, IMA Journal of Mathematical Control and Information, 23 (2006), 85. Google Scholar

[48]

G. Q. Xu, D. Y. Liu and Y. Q. Liu, Abstract second order hyperbolic system and applications to controlled networks of strings,, SIAM J. Control Optim., 47 (2008), 1762. doi: 10.1137/060649367. Google Scholar

[49]

G. Q. Xu and S. P. Yung, The expansion of semigroup and criterion of Riesz basis,, Journal of Differential Equations, 210 (2005), 1. doi: 10.1016/j.jde.2004.09.015. Google Scholar

[50]

G. Q. Xu, S. P. Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control,, ESAIM: Control, 12 (2006), 770. doi: 10.1051/cocv:2006021. Google Scholar

[51]

R. M. Young, "An Introduction to Nonharmonic Fourier Series,", Pure and Applied Mathematics, 93 (1980). Google Scholar

show all references

References:
[1]

R. A. Adams, "Sobolev Spaces,", Pure and Applied Mathematics, 65 (1975). Google Scholar

[2]

K. Ammari, Asymptotic behaviour of some elastic planar networks of Bernoulli-Euler beams,, Appl. Anal., 86 (2007), 1529. doi: 10.1080/00036810701734113. Google Scholar

[3]

K. Ammari and M. Jellouli, Stabilization of star-shaped networks of strings,, Differential and Integral Equations, 17 (2004), 1395. Google Scholar

[4]

K. Ammari and M. Jellouli, Remark on stabilization of tree-shaped networks of strings,, Applications of Mathematics, 52 (2007), 327. doi: 10.1007/s10492-007-0018-1. Google Scholar

[5]

K. Ammari, M. Jellouli and M. Khenissi, Stabilization of generic trees of strings,, Journal of Dynamical and Control Systems, 11 (2005), 177. doi: 10.1007/s10883-005-4169-7. Google Scholar

[6]

S. A. Avdonin and S. A. Ivanov, "Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems,", Cambridge University Press, (1995). Google Scholar

[7]

J. W. Brown and R. V. Churchill, "Complex Variables and Applications," Seventh Edition,, China Machine Press, (2004). Google Scholar

[8]

P. G. Casazza and G. Kutyniok, Frames of subspaces,, Contemp. Math., 345 (2004), 87. Google Scholar

[9]

G. Chen, M. Coleman and H. H. West, Pointwise stabilization in the middle of the span for second order systems, nonuniform and uniform exponential decay of solutions,, SIAM J. Appl. Math., 47 (1987), 751. doi: 10.1137/0147052. Google Scholar

[10]

G. Chen, M. Delfour, A. Krall and G. Payre, Modeling, stabilization and control of seraially connected beams,, SIAM J. Control Optim, 25 (1987), 526. doi: 10.1137/0325029. Google Scholar

[11]

G. Chen, S. G. Krantz, D. L. Russell, C. E. Wayne, H. H. West and M. P. Coleman, Analysis, designs, and behavior of dissipative joints for coupled beams,, SIAM J. Appl. Math., 49 (1989), 1665. doi: 10.1137/0149101. Google Scholar

[12]

R. Datko, Two examples of ill-posedness with respect to small time delays in stabilized elastic systems,, IEEE Trans. Automatic Control, 38 (1993), 163. doi: 10.1109/9.186332. Google Scholar

[13]

R. Datko, J. Lagnese and M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations,, SIAM J. Control Optim., 24 (1986), 152. doi: 10.1137/0324007. Google Scholar

[14]

I. C. Gohberg and M. G. Krein, "Introduction to the Theory of Linear Nonselfadjoint Operators,", AMS Transl. Math. Monographs, (1969). Google Scholar

[15]

B. Z. Guo and Y. Xie, A sufficient condition on Riesz basis with parentheses of non-self-adjoint operator and application to a serially connected string system under joint feedbacks,, SIAM J. Control Optim., 43 (2004), 1234. doi: 10.1137/S0363012902420352. Google Scholar

[16]

B. Z. Guo and K. Y. Yang, Output feedback stabilization of a one-dimensional Schrödinger equation by boundary observation with time delay,, IEEE Transactions on Automatic Control, 55 (2010), 1226. doi: 10.1109/TAC.2010.2051070. Google Scholar

[17]

Z. J. Han and L. Wang, Riesz basis property and stability of planar networks of controlled strings,, Acta Appl. Math., 110 (2010), 511. doi: 10.1007/s10440-009-9459-8. Google Scholar

[18]

Z. J. Han and G. Q. Xu, Spectrum and dynamical behavior of a kind of planar network of non-uniform strings with non-collocated feedbacks,, Networks and Heterogeneous Media, 5 (2010), 315. doi: 10.3934/nhm.2010.5.315. Google Scholar

[19]

Z. J. Han and G. Q. Xu, Exponential stabilisation of a simple tree-shaped network of Timoshenko beams system,, International Journal of Control, 83 (2010), 1485. doi: 10.1080/00207179.2010.481767. Google Scholar

[20]

Z. J. Han, G. Q. Xu, Stabilization and Riesz basis of a star-shaped network of Timoshenko beams,, Journal of Dynamical and Control Systems, 16 (2010), 227. doi: 10.1007/s10883-010-9091-y. Google Scholar

[21]

Z. J. Han and G. Q. Xu, Stabilization and Riesz basis property of two serially connected Timoshenko beams system,, Z. Angew. Math. Mech., 89 (2009), 962. doi: 10.1002/zamm.200800176. Google Scholar

[22]

Z. J. Han and G. Q. Xu, Exponential stability of Timoshenko beam system with delay terms in boundary feedbacks,, ESAIM: Control, 17 (2011), 552. doi: 10.1051/cocv/2010009. Google Scholar

[23]

J. Lagnese, G. Leugering and E. J. P. G. Schmidt, "Modeling, Analysis of Dynamic Elastic Multi-Link Structures,", Birkhäuser-Verlag, (1994). Google Scholar

[24]

J. S. Liang and Y. Q. Chen, Boundary control of wave equations with delayed boundary measurement,, Proceedings of IEEE International Conference on Robotics and Biomimetics, (2004), 849. doi: 10.1109/ROBIO.2004.1521895. Google Scholar

[25]

J. S. Liang, Y. Q. Chen and B. Z. Guo, A new boundary control method for beam equation with delayed boundary measurement using modified smith predictors,, Proceedings of the 42nd IEEE Conference on Decision and Control, (2003), 809. Google Scholar

[26]

Yu. I. Lyubich and V. Q. Phóng, Asymptotic stability of linear differential equations in Banach spaces,, Studia Math., 88 (1988), 34. Google Scholar

[27]

R. Mennicken and M. Möller, "Non-self-adjoint Boundary Eigenvalue Problem,", North-Holland Mathematics Studies, 192 (2003). Google Scholar

[28]

D. Mercier, Spectrum analysis of a serially connected Euler-Bernoulli beams problems,, Networks and Heterogeneous Media, 4 (2009), 709. doi: 10.3934/nhm.2009.4.709. Google Scholar

[29]

D. Mercier and V. Régnier, Spectrum of a network of Euler-Bernoulli beams,, Journal of Mathematical Analysis and Applications, 337 (2008), 174. doi: 10.1016/j.jmaa.2007.03.080. Google Scholar

[30]

D. Mercier and V. Régnier, Control of a network of Euler-Bernoulli beams,, Journal of Mathematical Analysis and Applications, 342 (2008), 874. doi: 10.1016/j.jmaa.2007.12.062. Google Scholar

[31]

W. Michiels and S. I. Niculescu, "Stability and Stabilization of Time-Delay Systems. An Eigenvalue-Based Approach,", Society for Industrial and Applied Mathematics, (2007). doi: 10.1137/1.9780898718645. Google Scholar

[32]

O. Morgul, On the stabilization and stability robustness against small delays of some damped wave equation,, IEEE Trans. Automatic Control, 40 (1995), 1626. doi: 10.1109/9.412634. Google Scholar

[33]

S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks,, SIAM J. Control Optim., 45 (2006), 1561. doi: 10.1137/060648891. Google Scholar

[34]

S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay,, Differential and Integral Equations, 21 (2008), 935. Google Scholar

[35]

S. Nicaise and J. Valein, Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks,, Networks and Heterogeneous Media, 2 (2007), 425. doi: 10.3934/nhm.2007.2.425. Google Scholar

[36]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Springer-Verlag, (1983). Google Scholar

[37]

A. A. Shkalikov, Boundary problems for ordinary differential equations with parameter in the boundary conditions,, J. Soviet Math., 33 (1986), 1311. doi: 10.1007/BF01084754. Google Scholar

[38]

K. Sriram and M. S. Gopinathan, A two variable delay model for the circadian rhythm of Neurospora crassa,, J. Theor. Biol., 231 (2004), 23. doi: 10.1016/j.jtbi.2004.04.006. Google Scholar

[39]

J. Srividhya and M. S. Gopinathan, A simple time delay model for eukaryotic cell cycle,, Journal of Theoretical Biology, 241 (2006), 617. doi: 10.1016/j.jtbi.2005.12.020. Google Scholar

[40]

H. Suh and Z. Bien, Use of time-delay actions in the controller design,, IEEE Trans. Automatic Control, 25 (1980), 600. doi: 10.1109/TAC.1980.1102347. Google Scholar

[41]

S. Timoshenko, "Vibration Problems in Engineering,", Van Norstrand, (1955). Google Scholar

[42]

J. Valein and E. Zuazua, Stabilization of the wave equation on 1-d networks,, SIAM J. Contr. Optim, 48 (2009), 2771. doi: 10.1137/080733590. Google Scholar

[43]

Q. P. Vu, J. M. Wang, G. Q. Xu and S. P. Yung, Spectral analysis and system of fundamental solutions for Timoshenko beams,, Appl. Math. Lett., 18 (2005), 127. doi: 10.1016/j.aml.2004.09.001. Google Scholar

[44]

J. M. Wang and B. Z. Guo, Riesz basis and stabilization for the flexible structure of a symmetric tree-shaped beam network,, Math. Meth. Appl. Sci., 31 (2008), 289. doi: 10.1002/mma.909. Google Scholar

[45]

G. Q. Xu, B. Z. Guo, Riesz basis property of evolution equations in Hilbert spaces and application to a coupled string equation,, SIAM J. Control Optim., 42 (2003), 966. doi: 10.1137/S0363012901400081. Google Scholar

[46]

G. Q. Xu, Z. J. Han and S. P. Yung, Riesz basis property of serially connected Timoshenko beams,, International Journal of Control, 80 (2007), 470. doi: 10.1080/00207170601100904. Google Scholar

[47]

G. Q. Xu and J. G. Jia, The group and Riesz basis properties of string systems with time delay and exact controllability with boundary control,, IMA Journal of Mathematical Control and Information, 23 (2006), 85. Google Scholar

[48]

G. Q. Xu, D. Y. Liu and Y. Q. Liu, Abstract second order hyperbolic system and applications to controlled networks of strings,, SIAM J. Control Optim., 47 (2008), 1762. doi: 10.1137/060649367. Google Scholar

[49]

G. Q. Xu and S. P. Yung, The expansion of semigroup and criterion of Riesz basis,, Journal of Differential Equations, 210 (2005), 1. doi: 10.1016/j.jde.2004.09.015. Google Scholar

[50]

G. Q. Xu, S. P. Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control,, ESAIM: Control, 12 (2006), 770. doi: 10.1051/cocv:2006021. Google Scholar

[51]

R. M. Young, "An Introduction to Nonharmonic Fourier Series,", Pure and Applied Mathematics, 93 (1980). Google Scholar

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