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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 & 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.  Google Scholar

[2]

M. Aouadi, Stability aspects in a nonsimple thermoelastic diffusion problem, Appl. Anal., 92 (2013), 1816-1828. doi: 10.1080/00036811.2012.702341.  Google Scholar

[3]

M. Aouadi and T. Moulahi, Asymptotic analysis of a nonsimple thermoelastic rod, Accepted in Disc. Cont. Dyn. Syst., (2015). Google Scholar

[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.  Google Scholar

[5]

G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation, Rend. Istit. Mat. Univ. Trieste, 28 (1996), 1-28.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems, Lecture Notes in Control and Information Sciences, Springer, Berlin, 1978.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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S. W. Hansen, Boundary control of a one-dimensional linear thermoelastic rod, SIAM J. Control Optim., 32 (1994), 1052-1074. doi: 10.1137/S0363012991222607.  Google Scholar

[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.  Google Scholar

[16]

D. Ieşan, Thermoelastic Models of Continua, Kluwer Academic Publishers, Dordrecht, 2004. doi: 10.1007/978-1-4020-2310-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

J.-L. Lions, Contrôlabilité, Exacte Perturbations et Stabilisation de Systèmes Distribués, Tome 1, Rech. Math. Appl. 8, Masson, Paris, 1988.  Google Scholar

[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.  Google Scholar

[25]

K. Narukawa, Boundary value control of thermoelastic systems, Hiroshima Math. J., 13 (1983), 227-272.  Google Scholar

[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.  Google Scholar

[27]

R. Quintanilla, Thermoelasticity without energy dissipation of nonsimple materials, Z. Angew. Math. Mech., 83 (2003), 172-180. doi: 10.1002/zamm.200310017.  Google Scholar

[28]

A. E. Taylor and D. C. Lay, Introduction to Functional Analysis, John Wiley and Sons. New York. 1980.  Google Scholar

[29]

L. de Teresa and E. Zuazua, Controllability for the linear system of thermoelastic plates, Adv. Diff. Equat., 1 (1996), 369-402.  Google Scholar

[30]

E. Zuazua, Controllability of the linear system of thermoelasticity, J. Math. Pures. Appl., 74 (1995), 291-315.  Google Scholar

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.  Google Scholar

[2]

M. Aouadi, Stability aspects in a nonsimple thermoelastic diffusion problem, Appl. Anal., 92 (2013), 1816-1828. doi: 10.1080/00036811.2012.702341.  Google Scholar

[3]

M. Aouadi and T. Moulahi, Asymptotic analysis of a nonsimple thermoelastic rod, Accepted in Disc. Cont. Dyn. Syst., (2015). Google Scholar

[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.  Google Scholar

[5]

G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation, Rend. Istit. Mat. Univ. Trieste, 28 (1996), 1-28.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems, Lecture Notes in Control and Information Sciences, Springer, Berlin, 1978.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

D. Ieşan, Thermoelastic Models of Continua, Kluwer Academic Publishers, Dordrecht, 2004. doi: 10.1007/978-1-4020-2310-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

J.-L. Lions, Contrôlabilité, Exacte Perturbations et Stabilisation de Systèmes Distribués, Tome 1, Rech. Math. Appl. 8, Masson, Paris, 1988.  Google Scholar

[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.  Google Scholar

[25]

K. Narukawa, Boundary value control of thermoelastic systems, Hiroshima Math. J., 13 (1983), 227-272.  Google Scholar

[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.  Google Scholar

[27]

R. Quintanilla, Thermoelasticity without energy dissipation of nonsimple materials, Z. Angew. Math. Mech., 83 (2003), 172-180. doi: 10.1002/zamm.200310017.  Google Scholar

[28]

A. E. Taylor and D. C. Lay, Introduction to Functional Analysis, John Wiley and Sons. New York. 1980.  Google Scholar

[29]

L. de Teresa and E. Zuazua, Controllability for the linear system of thermoelastic plates, Adv. Diff. Equat., 1 (1996), 369-402.  Google Scholar

[30]

E. Zuazua, Controllability of the linear system of thermoelasticity, J. Math. Pures. Appl., 74 (1995), 291-315.  Google Scholar

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