American Institute of Mathematical Sciences

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June  2015, 5(2): 321-333. doi: 10.3934/mcrf.2015.5.321

Exponential stability of a joint-leg-beam system with memory damping

Qiong Zhang 1,

1. 

School of Mathematics, Beijing Institute of Technology, Beijing, 100081, China

Received  March 2014 Revised  July 2014 Published  April 2015

In this paper, we consider a system for combined axial and transverse motions of two viscoelastic Euler-Bernoulli beams connected through two legs to a joint. This model comes from rigidizable and inflatable space structures. First, the exponential stability of the joint-leg-beam system is obtained when both beams are subject to viscoelastic damping and memory kernels satisfy reasonable assumptions. Then, we show the lack of uniform decay of the coupled system when only one beam is assumed to have a memory damping and the second beam has no damping.
Keywords: Joint-leg-beam system, memory damping, exponential stability., $C_0$-semigroup.
Mathematics Subject Classification: Primary: 37B35, 35B4.
Citation: Qiong Zhang. Exponential stability of a joint-leg-beam system with memory damping. Mathematical Control & Related Fields, 2015, 5 (2) : 321-333. doi: 10.3934/mcrf.2015.5.321
References:
[1]

C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM Journal on Control and Optimization, 30 (1992), 1024-1065. doi: 10.1137/0330055.  Google Scholar

[2]

C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, Journal of Evolution Equations, 8 (2008), 765-780. doi: 10.1007/s00028-008-0424-1.  Google Scholar

[3]

J. A. Burns, E. M. Cliff, Z. Liu and R. D. Spies, On coupled transversal and axial motions of two beams with a joint, Journal of Mathematical Analysis and Applications, 339 (2008), 182-196. doi: 10.1016/j.jmaa.2007.06.047.  Google Scholar

[4]

J. A. Burns, E. M. Cliff, Z. Liu and R. D. Spies, Polynomial stability of a joint-leg-beam system with local damping, Mathematical and Computer Modelling, 46 (2007), 1236-1246. doi: 10.1016/j.mcm.2006.11.037.  Google Scholar

[5]

V. V. Chepyzhov and V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity, Asymptotic Analysis, 46 (2006), 251-273.  Google Scholar

[6]

E. M. Cliff, B. Fulton, T. Herdman, Z. Liu and R. D. Spies, Well posedness and exponential stability of a thermoelastic joint-leg-beam system with Robin boundary conditions, Mathematical and Computer Modelling, 49 (2009), 1097-1108. doi: 10.1016/j.mcm.2008.03.018.  Google Scholar

[7]

M. Fabrizio, C. Giorgi and V. Pata, A new approach to equations with memory, Archive for Rational Mechanics and Analysis, 198 (2010), 189-232. doi: 10.1007/s00205-010-0300-3.  Google Scholar

[8]

K. Guidanean and D. Lichodziejewski, An Inflatable Rigidizable Truss Structure Based on new Sub-Tg Polyurethane Composites, AIAA Paper 02-1593, AIAA Publications, 2002. doi: 10.2514/6.2002-1593.  Google Scholar

[9]

F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Annals of Differential Equations, 1 (1985), 43-56.  Google Scholar

[10]

F. L. Huang, Strong asymptotic stability of linear dynamical systems in Banach spaces, Journal of Differential Equations, 104 (1993), 307-324. doi: 10.1006/jdeq.1993.1074.  Google Scholar

[11]

C. H. M. Jenkins, ed., Gossamer Spacecraft: Membrane and Inflatable Technology for Space Applications, AIAA Progress in Aeronautics and Astronautics, 191, 2001. Google Scholar

[12]

J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modeling, Analysis and Control of Dynamic Elastic Multi-Link Structures, Birkhäuser, Boston, 1994. doi: 10.1007/978-1-4612-0273-8.  Google Scholar

[13]

K. Liu and Z. Liu, Exponential decay of energy of vibrating strings with local viscoelasticity, Zeitschrift für angewandte Mathematik und Physik, 53 (2002), 265-280. doi: 10.1007/s00033-002-8155-6.  Google Scholar

[14]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, NY, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[15]

J. Prüss, On the spectrum of $C_0-$semigroups, Transactions of the American Mathematical Society, 284 (1984), 847-857. doi: 10.2307/1999112.  Google Scholar

[16]

B. Rao, Stabilization of elastic plates with dynamical boundary control, SIAM Journal on Control and Optimization, 36 (1998), 148-163. doi: 10.1137/S0363012996300975.  Google Scholar

[17]

Q. Zhang, Stability analysis of an interactive system of wave equation and heat equation with memory, Z. Angew. Math. Phys., 65 (2014), 905-923. doi: 10.1007/s00033-013-0366-5.  Google Scholar

show all references

References:
[1]

C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM Journal on Control and Optimization, 30 (1992), 1024-1065. doi: 10.1137/0330055.  Google Scholar

[2]

C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, Journal of Evolution Equations, 8 (2008), 765-780. doi: 10.1007/s00028-008-0424-1.  Google Scholar

[3]

J. A. Burns, E. M. Cliff, Z. Liu and R. D. Spies, On coupled transversal and axial motions of two beams with a joint, Journal of Mathematical Analysis and Applications, 339 (2008), 182-196. doi: 10.1016/j.jmaa.2007.06.047.  Google Scholar

[4]

J. A. Burns, E. M. Cliff, Z. Liu and R. D. Spies, Polynomial stability of a joint-leg-beam system with local damping, Mathematical and Computer Modelling, 46 (2007), 1236-1246. doi: 10.1016/j.mcm.2006.11.037.  Google Scholar

[5]

V. V. Chepyzhov and V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity, Asymptotic Analysis, 46 (2006), 251-273.  Google Scholar

[6]

E. M. Cliff, B. Fulton, T. Herdman, Z. Liu and R. D. Spies, Well posedness and exponential stability of a thermoelastic joint-leg-beam system with Robin boundary conditions, Mathematical and Computer Modelling, 49 (2009), 1097-1108. doi: 10.1016/j.mcm.2008.03.018.  Google Scholar

[7]

M. Fabrizio, C. Giorgi and V. Pata, A new approach to equations with memory, Archive for Rational Mechanics and Analysis, 198 (2010), 189-232. doi: 10.1007/s00205-010-0300-3.  Google Scholar

[8]

K. Guidanean and D. Lichodziejewski, An Inflatable Rigidizable Truss Structure Based on new Sub-Tg Polyurethane Composites, AIAA Paper 02-1593, AIAA Publications, 2002. doi: 10.2514/6.2002-1593.  Google Scholar

[9]

F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Annals of Differential Equations, 1 (1985), 43-56.  Google Scholar

[10]

F. L. Huang, Strong asymptotic stability of linear dynamical systems in Banach spaces, Journal of Differential Equations, 104 (1993), 307-324. doi: 10.1006/jdeq.1993.1074.  Google Scholar

[11]

C. H. M. Jenkins, ed., Gossamer Spacecraft: Membrane and Inflatable Technology for Space Applications, AIAA Progress in Aeronautics and Astronautics, 191, 2001. Google Scholar

[12]

J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modeling, Analysis and Control of Dynamic Elastic Multi-Link Structures, Birkhäuser, Boston, 1994. doi: 10.1007/978-1-4612-0273-8.  Google Scholar

[13]

K. Liu and Z. Liu, Exponential decay of energy of vibrating strings with local viscoelasticity, Zeitschrift für angewandte Mathematik und Physik, 53 (2002), 265-280. doi: 10.1007/s00033-002-8155-6.  Google Scholar

[14]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, NY, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[15]

J. Prüss, On the spectrum of $C_0-$semigroups, Transactions of the American Mathematical Society, 284 (1984), 847-857. doi: 10.2307/1999112.  Google Scholar

[16]

B. Rao, Stabilization of elastic plates with dynamical boundary control, SIAM Journal on Control and Optimization, 36 (1998), 148-163. doi: 10.1137/S0363012996300975.  Google Scholar

[17]

Q. Zhang, Stability analysis of an interactive system of wave equation and heat equation with memory, Z. Angew. Math. Phys., 65 (2014), 905-923. doi: 10.1007/s00033-013-0366-5.  Google Scholar

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