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Approximate controllability of abstract nonsimple thermoelastic problem
1. | Ecole Nationale d'Ingénieurs de Bizerte, Université de Carthage, BP66, Campus Universitaire Menzel Abderrahman 7035 |
2. | Faculté des Sciences de Bizerte, 7021 Zarzouna, Université de Carthage, Tunisia |
References:
[1] |
M. Aouadi, On uniform decay of a nonsimple thermoelastic bar with memory, J. Math. Analysis Appl., 402 (2013), 745-757.
doi: 10.1016/j.jmaa.2013.01.059. |
[2] |
M. Aouadi, Stability aspects in a nonsimple thermoelastic diffusion problem, Appl. Anal., 92 (2013), 1816-1828.
doi: 10.1080/00036811.2012.702341. |
[3] |
M. Aouadi and T. Moulahi, Asymptotic analysis of a nonsimple thermoelastic rod, Accepted in Disc. Cont. Dyn. Syst., (2015). |
[4] |
G. Avalos, Null controllability of von Karman thermoelastic plates under the clamped or free mechanical boundary conditions, J. Math. Anal. Appl., 318 (2006), 410-432.
doi: 10.1016/j.jmaa.2005.05.040. |
[5] |
G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation, Rend. Istit. Mat. Univ. Trieste, 28 (1996), 1-28. |
[6] |
G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system with free boundary conditions without mechanical dissipation, SIAM J. Math. Anal., 29 (1998), 155-182.
doi: 10.1137/S0036141096300823. |
[7] |
S. Axler, P. Bourdon and W. Ramey, Harmonic Function Theory, Graduate Texts in Mathematics, 137, Springer, New York, NY, USA, 1992.
doi: 10.1007/b97238. |
[8] |
A. Benabdallah and I. Lasiecka, Exponential decay rates for a full von Karman system of dynamic thermoelasticity, J. Diff. Equat., 160 (2000), 51-93.
doi: 10.1006/jdeq.1999.3656. |
[9] |
M. Ciarletta and D. Ieşan, Non-classical Elastic Solids, Pitman Research Notes in Mathematical Series, vol. 293, John Wiley & Sons, Inc., New York, 1993. |
[10] |
R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems, Lecture Notes in Control and Information Sciences, Springer, Berlin, 1978. |
[11] |
R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Texts in Applied Mathematics, 21, Springer, New York, NY, USA, 1995.
doi: 10.1007/978-1-4612-4224-6. |
[12] |
H. D. Fernàndez Sare, J. E. Munoz Rivera and R. Quintanilla, Decay of solutions in nonsimple thermoelastic bars, Int. J. Eng. Sci., 48 (2010), 1233-1241.
doi: 10.1016/j.ijengsci.2010.04.014. |
[13] |
J. A. Gawinecki and J. Lazuka, Global solution on Cauchy problem in nonlinear non-simple thermoelastic materials, Proc. Appl. Math. Mech., 6 (2006), 371-372.
doi: 10.1002/pamm.200610167. |
[14] |
S. W. Hansen, Boundary control of a one-dimensional linear thermoelastic rod, SIAM J. Control Optim., 32 (1994), 1052-1074.
doi: 10.1137/S0363012991222607. |
[15] |
S. W. Hansen, Exponential energy decay in a linear thermoelastic rod, J. Math. Anal. Appl., 167 (1992), 429-442.
doi: 10.1016/0022-247X(92)90217-2. |
[16] |
D. Ieşan, Thermoelastic Models of Continua, Kluwer Academic Publishers, Dordrecht, 2004.
doi: 10.1007/978-1-4020-2310-1. |
[17] |
H. Kolakowski and J. Lazuka, The Cauchy problem for the system of partial differential equations describing nonsimple thermoelasticity, Appl. Math., 35 (2008), 97-105.
doi: 10.4064/am35-1-6. |
[18] |
I. Lasiecka and R. Triggiani, Two direct proofs on the analyticity of the s.c. semigroup arising in abstract thermo-elastic equations, Adv. Diff. Equat., 3 (1998), 387-416. |
[19] |
G. Lebeau and E. Zuazua, Null controllability of a system of linear thermoelasticity, Archives Rat. Mech. Anal., 141 (1998), 297-329.
doi: 10.1007/s002050050078. |
[20] |
H. Leiva, Existence of bounded solutions of a second order system with dissipation, J. Math. Anal. Appl., 237 (1999), 288-302.
doi: 10.1006/jmaa.1999.6480. |
[21] |
H. Leiva and H. Zambrano, Rank condition for the controllability of a linear time-varying system, Int. J. Control, 72 (1999), 929-931.
doi: 10.1080/002071799220669. |
[22] |
H. Leiva, A necessary and sufficient algebraic condition for the controllability of a thermoelastic plate equation, IMA J. Math. Cont. Inf., 20 (2003), 393-410.
doi: 10.1093/imamci/20.4.393. |
[23] |
J.-L. Lions, Contrôlabilité, Exacte Perturbations et Stabilisation de Systèmes Distribués, Tome 1, Rech. Math. Appl. 8, Masson, Paris, 1988. |
[24] |
W. J. Liu, Partial exact controllability and exponential stability of the higher dimensional linear thermoelasticity, ESAIM Contrôle Optim. C-alc. Var., 3 (1998), 23-48.
doi: 10.1051/cocv:1998101. |
[25] |
K. Narukawa, Boundary value control of thermoelastic systems, Hiroshima Math. J., 13 (1983), 227-272. |
[26] |
V. Pata and R. Quintanilla, On the decay of solutions in nonsimple elastic solids with memory, J. Math. Anal. Appl., 363 (2010), 19-28.
doi: 10.1016/j.jmaa.2009.07.055. |
[27] |
R. Quintanilla, Thermoelasticity without energy dissipation of nonsimple materials, Z. Angew. Math. Mech., 83 (2003), 172-180.
doi: 10.1002/zamm.200310017. |
[28] |
A. E. Taylor and D. C. Lay, Introduction to Functional Analysis, John Wiley and Sons. New York. 1980. |
[29] |
L. de Teresa and E. Zuazua, Controllability for the linear system of thermoelastic plates, Adv. Diff. Equat., 1 (1996), 369-402. |
[30] |
E. Zuazua, Controllability of the linear system of thermoelasticity, J. Math. Pures. Appl., 74 (1995), 291-315. |
show all references
References:
[1] |
M. Aouadi, On uniform decay of a nonsimple thermoelastic bar with memory, J. Math. Analysis Appl., 402 (2013), 745-757.
doi: 10.1016/j.jmaa.2013.01.059. |
[2] |
M. Aouadi, Stability aspects in a nonsimple thermoelastic diffusion problem, Appl. Anal., 92 (2013), 1816-1828.
doi: 10.1080/00036811.2012.702341. |
[3] |
M. Aouadi and T. Moulahi, Asymptotic analysis of a nonsimple thermoelastic rod, Accepted in Disc. Cont. Dyn. Syst., (2015). |
[4] |
G. Avalos, Null controllability of von Karman thermoelastic plates under the clamped or free mechanical boundary conditions, J. Math. Anal. Appl., 318 (2006), 410-432.
doi: 10.1016/j.jmaa.2005.05.040. |
[5] |
G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation, Rend. Istit. Mat. Univ. Trieste, 28 (1996), 1-28. |
[6] |
G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system with free boundary conditions without mechanical dissipation, SIAM J. Math. Anal., 29 (1998), 155-182.
doi: 10.1137/S0036141096300823. |
[7] |
S. Axler, P. Bourdon and W. Ramey, Harmonic Function Theory, Graduate Texts in Mathematics, 137, Springer, New York, NY, USA, 1992.
doi: 10.1007/b97238. |
[8] |
A. Benabdallah and I. Lasiecka, Exponential decay rates for a full von Karman system of dynamic thermoelasticity, J. Diff. Equat., 160 (2000), 51-93.
doi: 10.1006/jdeq.1999.3656. |
[9] |
M. Ciarletta and D. Ieşan, Non-classical Elastic Solids, Pitman Research Notes in Mathematical Series, vol. 293, John Wiley & Sons, Inc., New York, 1993. |
[10] |
R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems, Lecture Notes in Control and Information Sciences, Springer, Berlin, 1978. |
[11] |
R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Texts in Applied Mathematics, 21, Springer, New York, NY, USA, 1995.
doi: 10.1007/978-1-4612-4224-6. |
[12] |
H. D. Fernàndez Sare, J. E. Munoz Rivera and R. Quintanilla, Decay of solutions in nonsimple thermoelastic bars, Int. J. Eng. Sci., 48 (2010), 1233-1241.
doi: 10.1016/j.ijengsci.2010.04.014. |
[13] |
J. A. Gawinecki and J. Lazuka, Global solution on Cauchy problem in nonlinear non-simple thermoelastic materials, Proc. Appl. Math. Mech., 6 (2006), 371-372.
doi: 10.1002/pamm.200610167. |
[14] |
S. W. Hansen, Boundary control of a one-dimensional linear thermoelastic rod, SIAM J. Control Optim., 32 (1994), 1052-1074.
doi: 10.1137/S0363012991222607. |
[15] |
S. W. Hansen, Exponential energy decay in a linear thermoelastic rod, J. Math. Anal. Appl., 167 (1992), 429-442.
doi: 10.1016/0022-247X(92)90217-2. |
[16] |
D. Ieşan, Thermoelastic Models of Continua, Kluwer Academic Publishers, Dordrecht, 2004.
doi: 10.1007/978-1-4020-2310-1. |
[17] |
H. Kolakowski and J. Lazuka, The Cauchy problem for the system of partial differential equations describing nonsimple thermoelasticity, Appl. Math., 35 (2008), 97-105.
doi: 10.4064/am35-1-6. |
[18] |
I. Lasiecka and R. Triggiani, Two direct proofs on the analyticity of the s.c. semigroup arising in abstract thermo-elastic equations, Adv. Diff. Equat., 3 (1998), 387-416. |
[19] |
G. Lebeau and E. Zuazua, Null controllability of a system of linear thermoelasticity, Archives Rat. Mech. Anal., 141 (1998), 297-329.
doi: 10.1007/s002050050078. |
[20] |
H. Leiva, Existence of bounded solutions of a second order system with dissipation, J. Math. Anal. Appl., 237 (1999), 288-302.
doi: 10.1006/jmaa.1999.6480. |
[21] |
H. Leiva and H. Zambrano, Rank condition for the controllability of a linear time-varying system, Int. J. Control, 72 (1999), 929-931.
doi: 10.1080/002071799220669. |
[22] |
H. Leiva, A necessary and sufficient algebraic condition for the controllability of a thermoelastic plate equation, IMA J. Math. Cont. Inf., 20 (2003), 393-410.
doi: 10.1093/imamci/20.4.393. |
[23] |
J.-L. Lions, Contrôlabilité, Exacte Perturbations et Stabilisation de Systèmes Distribués, Tome 1, Rech. Math. Appl. 8, Masson, Paris, 1988. |
[24] |
W. J. Liu, Partial exact controllability and exponential stability of the higher dimensional linear thermoelasticity, ESAIM Contrôle Optim. C-alc. Var., 3 (1998), 23-48.
doi: 10.1051/cocv:1998101. |
[25] |
K. Narukawa, Boundary value control of thermoelastic systems, Hiroshima Math. J., 13 (1983), 227-272. |
[26] |
V. Pata and R. Quintanilla, On the decay of solutions in nonsimple elastic solids with memory, J. Math. Anal. Appl., 363 (2010), 19-28.
doi: 10.1016/j.jmaa.2009.07.055. |
[27] |
R. Quintanilla, Thermoelasticity without energy dissipation of nonsimple materials, Z. Angew. Math. Mech., 83 (2003), 172-180.
doi: 10.1002/zamm.200310017. |
[28] |
A. E. Taylor and D. C. Lay, Introduction to Functional Analysis, John Wiley and Sons. New York. 1980. |
[29] |
L. de Teresa and E. Zuazua, Controllability for the linear system of thermoelastic plates, Adv. Diff. Equat., 1 (1996), 369-402. |
[30] |
E. Zuazua, Controllability of the linear system of thermoelasticity, J. Math. Pures. Appl., 74 (1995), 291-315. |
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