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Stability and controllability of a wave equation with dynamical boundary control
Exponential stability of a joint-leg-beam system with memory damping
1. | School of Mathematics, Beijing Institute of Technology, Beijing, 100081, China |
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. |
[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. |
[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. |
[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. |
[5] |
V. V. Chepyzhov and V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity, Asymptotic Analysis, 46 (2006), 251-273. |
[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. |
[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. |
[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. |
[9] |
F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Annals of Differential Equations, 1 (1985), 43-56. |
[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. |
[11] |
C. H. M. Jenkins, ed., Gossamer Spacecraft: Membrane and Inflatable Technology for Space Applications, AIAA Progress in Aeronautics and Astronautics, 191, 2001. |
[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. |
[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. |
[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. |
[15] |
J. Prüss, On the spectrum of $C_0-$semigroups, Transactions of the American Mathematical Society, 284 (1984), 847-857.
doi: 10.2307/1999112. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[5] |
V. V. Chepyzhov and V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity, Asymptotic Analysis, 46 (2006), 251-273. |
[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. |
[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. |
[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. |
[9] |
F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Annals of Differential Equations, 1 (1985), 43-56. |
[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. |
[11] |
C. H. M. Jenkins, ed., Gossamer Spacecraft: Membrane and Inflatable Technology for Space Applications, AIAA Progress in Aeronautics and Astronautics, 191, 2001. |
[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. |
[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. |
[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. |
[15] |
J. Prüss, On the spectrum of $C_0-$semigroups, Transactions of the American Mathematical Society, 284 (1984), 847-857.
doi: 10.2307/1999112. |
[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. |
[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. |
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