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December  2015, 4(4): 373-389. doi: 10.3934/eect.2015.4.373

Approximate controllability of abstract nonsimple thermoelastic problem

Moncef Aouadi 1, and  Taoufik Moulahi 2,

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

Received  April 2015 Revised  October 2015 Published  November 2015

In this paper, an abstract nonsimple thermoelastic problem involving higher order gradients of displacement is considered with Dirichlet boundary conditions. We prove that the linear operator of the proposed system generates a strongly continuous semigroup which decays exponentially to zero. The optimal decay rate is determined explicitly by the physical parameters of the problem. Then we show the approximate controllability of the considered problem.
Keywords: semigroup, Nonsimple thermoelasticity, approximate controllability., exponential decay, optimal decay rate.
Mathematics Subject Classification: Primary: 49K20; Secondary: 35B3.
Citation: Moncef Aouadi, Taoufik Moulahi. Approximate controllability of abstract nonsimple thermoelastic problem. Evolution Equations and Control Theory, 2015, 4 (4) : 373-389. doi: 10.3934/eect.2015.4.373
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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