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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 and 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, Vol. 65, Academic Press, New York, 1975.

[2]

K. Ammari, Asymptotic behaviour of some elastic planar networks of Bernoulli-Euler beams, Appl. Anal., 86 (2007), 1529-1548. doi: 10.1080/00036810701734113.

[3]

K. Ammari and M. Jellouli, Stabilization of star-shaped networks of strings, Differential and Integral Equations, 17 (2004), 1395-1410.

[4]

K. Ammari and M. Jellouli, Remark on stabilization of tree-shaped networks of strings, Applications of Mathematics, 52 (2007), 327-343. doi: 10.1007/s10492-007-0018-1.

[5]

K. Ammari, M. Jellouli and M. Khenissi, Stabilization of generic trees of strings, Journal of Dynamical and Control Systems, 11 (2005), 177-193. doi: 10.1007/s10883-005-4169-7.

[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, Cambridge, 1995.

[7]

J. W. Brown and R. V. Churchill, "Complex Variables and Applications," Seventh Edition, China Machine Press, Beijing, 2004.

[8]

P. G. Casazza and G. Kutyniok, Frames of subspaces, Contemp. Math., 345 (2004), 87-113.

[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), 751-780. doi: 10.1137/0147052.

[10]

G. Chen, M. Delfour, A. Krall and G. Payre, Modeling, stabilization and control of seraially connected beams, SIAM J. Control Optim, 25 (1987), 526-546. doi: 10.1137/0325029.

[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), 1665-1693. doi: 10.1137/0149101.

[12]

R. Datko, Two examples of ill-posedness with respect to small time delays in stabilized elastic systems, IEEE Trans. Automatic Control, 38 (1993), 163-166. doi: 10.1109/9.186332.

[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), 152-156. doi: 10.1137/0324007.

[14]

I. C. Gohberg and M. G. Krein, "Introduction to the Theory of Linear Nonselfadjoint Operators," AMS Transl. Math. Monographs, American Mathematical Society, 1969.

[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), 1234-1252. doi: 10.1137/S0363012902420352.

[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), 1226-1232. doi: 10.1109/TAC.2010.2051070.

[17]

Z. J. Han and L. Wang, Riesz basis property and stability of planar networks of controlled strings, Acta Appl. Math., 110 (2010), 511-533. doi: 10.1007/s10440-009-9459-8.

[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), 315-334. doi: 10.3934/nhm.2010.5.315.

[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), 1485-1503. doi: 10.1080/00207179.2010.481767.

[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), 227-258. doi: 10.1007/s10883-010-9091-y.

[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), 962-980. doi: 10.1002/zamm.200800176.

[22]

Z. J. Han and G. Q. Xu, Exponential stability of Timoshenko beam system with delay terms in boundary feedbacks, ESAIM: Control, Optimisation and Calculus of Variations, 17 (2011), 552-574. doi: 10.1051/cocv/2010009.

[23]

J. Lagnese, G. Leugering and E. J. P. G. Schmidt, "Modeling, Analysis of Dynamic Elastic Multi-Link Structures," Birkhäuser-Verlag, Boston-Basel-Berlin, 1994.

[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, Shenyang, China, 849-854. doi: 10.1109/ROBIO.2004.1521895.

[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, Hawaii, USA, 809-814.

[26]

Yu. I. Lyubich and V. Q. Phóng, Asymptotic stability of linear differential equations in Banach spaces, Studia Math., 88 (1988), 34-37.

[27]

R. Mennicken and M. Möller, "Non-self-adjoint Boundary Eigenvalue Problem," North-Holland Mathematics Studies, vol. 192, North-Holland Publishing Co., Amsterdam, 2003.

[28]

D. Mercier, Spectrum analysis of a serially connected Euler-Bernoulli beams problems, Networks and Heterogeneous Media, 4 (2009), 709-730. doi: 10.3934/nhm.2009.4.709.

[29]

D. Mercier and V. Régnier, Spectrum of a network of Euler-Bernoulli beams, Journal of Mathematical Analysis and Applications, 337 (2008), 174-196. doi: 10.1016/j.jmaa.2007.03.080.

[30]

D. Mercier and V. Régnier, Control of a network of Euler-Bernoulli beams, Journal of Mathematical Analysis and Applications, 342 (2008), 874-894. doi: 10.1016/j.jmaa.2007.12.062.

[31]

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

[32]

O. Morgul, On the stabilization and stability robustness against small delays of some damped wave equation, IEEE Trans. Automatic Control, 40 (1995), 1626-1630. doi: 10.1109/9.412634.

[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), 1561-1585. doi: 10.1137/060648891.

[34]

S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay, Differential and Integral Equations, 21 (2008), 935-958.

[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), 425-479. doi: 10.3934/nhm.2007.2.425.

[36]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer-Verlag, Berlin, 1983.

[37]

A. A. Shkalikov, Boundary problems for ordinary differential equations with parameter in the boundary conditions, J. Soviet Math., 33 (1986), 1311-1342. doi: 10.1007/BF01084754.

[38]

K. Sriram and M. S. Gopinathan, A two variable delay model for the circadian rhythm of Neurospora crassa, J. Theor. Biol., 231 (2004), 23-38. doi: 10.1016/j.jtbi.2004.04.006.

[39]

J. Srividhya and M. S. Gopinathan, A simple time delay model for eukaryotic cell cycle, Journal of Theoretical Biology, 241 (2006), 617-627. doi: 10.1016/j.jtbi.2005.12.020.

[40]

H. Suh and Z. Bien, Use of time-delay actions in the controller design, IEEE Trans. Automatic Control, 25 (1980), 600-603. doi: 10.1109/TAC.1980.1102347.

[41]

S. Timoshenko, "Vibration Problems in Engineering," Van Norstrand, New York, 1955.

[42]

J. Valein and E. Zuazua, Stabilization of the wave equation on 1-d networks, SIAM J. Contr. Optim, 48 (2009), 2771-2797. doi: 10.1137/080733590.

[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), 127-134. doi: 10.1016/j.aml.2004.09.001.

[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), 289-314. doi: 10.1002/mma.909.

[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), 966-984. doi: 10.1137/S0363012901400081.

[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), 470-485. doi: 10.1080/00207170601100904.

[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), 85-96.

[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), 1762-1784. doi: 10.1137/060649367.

[49]

G. Q. Xu and S. P. Yung, The expansion of semigroup and criterion of Riesz basis, Journal of Differential Equations, 210 (2005), 1-24. doi: 10.1016/j.jde.2004.09.015.

[50]

G. Q. Xu, S. P. Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control, ESAIM: Control, Optimisation and Calculus of Variations, 12 (2006), 770-785. doi: 10.1051/cocv:2006021.

[51]

R. M. Young, "An Introduction to Nonharmonic Fourier Series," Pure and Applied Mathematics, vol. 93, Academic Press, London, 1980.

show all references

References:
[1]

R. A. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Vol. 65, Academic Press, New York, 1975.

[2]

K. Ammari, Asymptotic behaviour of some elastic planar networks of Bernoulli-Euler beams, Appl. Anal., 86 (2007), 1529-1548. doi: 10.1080/00036810701734113.

[3]

K. Ammari and M. Jellouli, Stabilization of star-shaped networks of strings, Differential and Integral Equations, 17 (2004), 1395-1410.

[4]

K. Ammari and M. Jellouli, Remark on stabilization of tree-shaped networks of strings, Applications of Mathematics, 52 (2007), 327-343. doi: 10.1007/s10492-007-0018-1.

[5]

K. Ammari, M. Jellouli and M. Khenissi, Stabilization of generic trees of strings, Journal of Dynamical and Control Systems, 11 (2005), 177-193. doi: 10.1007/s10883-005-4169-7.

[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, Cambridge, 1995.

[7]

J. W. Brown and R. V. Churchill, "Complex Variables and Applications," Seventh Edition, China Machine Press, Beijing, 2004.

[8]

P. G. Casazza and G. Kutyniok, Frames of subspaces, Contemp. Math., 345 (2004), 87-113.

[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), 751-780. doi: 10.1137/0147052.

[10]

G. Chen, M. Delfour, A. Krall and G. Payre, Modeling, stabilization and control of seraially connected beams, SIAM J. Control Optim, 25 (1987), 526-546. doi: 10.1137/0325029.

[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), 1665-1693. doi: 10.1137/0149101.

[12]

R. Datko, Two examples of ill-posedness with respect to small time delays in stabilized elastic systems, IEEE Trans. Automatic Control, 38 (1993), 163-166. doi: 10.1109/9.186332.

[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), 152-156. doi: 10.1137/0324007.

[14]

I. C. Gohberg and M. G. Krein, "Introduction to the Theory of Linear Nonselfadjoint Operators," AMS Transl. Math. Monographs, American Mathematical Society, 1969.

[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), 1234-1252. doi: 10.1137/S0363012902420352.

[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), 1226-1232. doi: 10.1109/TAC.2010.2051070.

[17]

Z. J. Han and L. Wang, Riesz basis property and stability of planar networks of controlled strings, Acta Appl. Math., 110 (2010), 511-533. doi: 10.1007/s10440-009-9459-8.

[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), 315-334. doi: 10.3934/nhm.2010.5.315.

[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), 1485-1503. doi: 10.1080/00207179.2010.481767.

[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), 227-258. doi: 10.1007/s10883-010-9091-y.

[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), 962-980. doi: 10.1002/zamm.200800176.

[22]

Z. J. Han and G. Q. Xu, Exponential stability of Timoshenko beam system with delay terms in boundary feedbacks, ESAIM: Control, Optimisation and Calculus of Variations, 17 (2011), 552-574. doi: 10.1051/cocv/2010009.

[23]

J. Lagnese, G. Leugering and E. J. P. G. Schmidt, "Modeling, Analysis of Dynamic Elastic Multi-Link Structures," Birkhäuser-Verlag, Boston-Basel-Berlin, 1994.

[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, Shenyang, China, 849-854. doi: 10.1109/ROBIO.2004.1521895.

[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, Hawaii, USA, 809-814.

[26]

Yu. I. Lyubich and V. Q. Phóng, Asymptotic stability of linear differential equations in Banach spaces, Studia Math., 88 (1988), 34-37.

[27]

R. Mennicken and M. Möller, "Non-self-adjoint Boundary Eigenvalue Problem," North-Holland Mathematics Studies, vol. 192, North-Holland Publishing Co., Amsterdam, 2003.

[28]

D. Mercier, Spectrum analysis of a serially connected Euler-Bernoulli beams problems, Networks and Heterogeneous Media, 4 (2009), 709-730. doi: 10.3934/nhm.2009.4.709.

[29]

D. Mercier and V. Régnier, Spectrum of a network of Euler-Bernoulli beams, Journal of Mathematical Analysis and Applications, 337 (2008), 174-196. doi: 10.1016/j.jmaa.2007.03.080.

[30]

D. Mercier and V. Régnier, Control of a network of Euler-Bernoulli beams, Journal of Mathematical Analysis and Applications, 342 (2008), 874-894. doi: 10.1016/j.jmaa.2007.12.062.

[31]

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

[32]

O. Morgul, On the stabilization and stability robustness against small delays of some damped wave equation, IEEE Trans. Automatic Control, 40 (1995), 1626-1630. doi: 10.1109/9.412634.

[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), 1561-1585. doi: 10.1137/060648891.

[34]

S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay, Differential and Integral Equations, 21 (2008), 935-958.

[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), 425-479. doi: 10.3934/nhm.2007.2.425.

[36]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer-Verlag, Berlin, 1983.

[37]

A. A. Shkalikov, Boundary problems for ordinary differential equations with parameter in the boundary conditions, J. Soviet Math., 33 (1986), 1311-1342. doi: 10.1007/BF01084754.

[38]

K. Sriram and M. S. Gopinathan, A two variable delay model for the circadian rhythm of Neurospora crassa, J. Theor. Biol., 231 (2004), 23-38. doi: 10.1016/j.jtbi.2004.04.006.

[39]

J. Srividhya and M. S. Gopinathan, A simple time delay model for eukaryotic cell cycle, Journal of Theoretical Biology, 241 (2006), 617-627. doi: 10.1016/j.jtbi.2005.12.020.

[40]

H. Suh and Z. Bien, Use of time-delay actions in the controller design, IEEE Trans. Automatic Control, 25 (1980), 600-603. doi: 10.1109/TAC.1980.1102347.

[41]

S. Timoshenko, "Vibration Problems in Engineering," Van Norstrand, New York, 1955.

[42]

J. Valein and E. Zuazua, Stabilization of the wave equation on 1-d networks, SIAM J. Contr. Optim, 48 (2009), 2771-2797. doi: 10.1137/080733590.

[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), 127-134. doi: 10.1016/j.aml.2004.09.001.

[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), 289-314. doi: 10.1002/mma.909.

[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), 966-984. doi: 10.1137/S0363012901400081.

[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), 470-485. doi: 10.1080/00207170601100904.

[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), 85-96.

[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), 1762-1784. doi: 10.1137/060649367.

[49]

G. Q. Xu and S. P. Yung, The expansion of semigroup and criterion of Riesz basis, Journal of Differential Equations, 210 (2005), 1-24. doi: 10.1016/j.jde.2004.09.015.

[50]

G. Q. Xu, S. P. Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control, ESAIM: Control, Optimisation and Calculus of Variations, 12 (2006), 770-785. doi: 10.1051/cocv:2006021.

[51]

R. M. Young, "An Introduction to Nonharmonic Fourier Series," Pure and Applied Mathematics, vol. 93, Academic Press, London, 1980.

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