March  2018, 7(1): 1-31. doi: 10.3934/eect.2018001

The controllability of a thermoelastic plate problem revisited

1. 

Université de Carthage, UR Systèmes dynamiques et applications, UR 17ES21, Ecole Nationale d'Ingénieurs de Bizerte, 7035, BP 66, Tunisia

2. 

Université de Carthage, UR Systèmes dynamiques et applications, UR 17ES21, Faculté des Sciences de Bizerte, Jarzouna 7021, Tunisia

Received  November 2016 Revised  November 2017 Published  January 2018

In this paper, the controllability for a thermoelastic plate problem with a rotational inertia parameter is considered under two scenarios. In the first case, we prove the exact and approximate controllability when the controls act in the whole domain. In the second case, we prove the interior approximate controllability when the controls act only on a subset of the domain. The distributed controls are determined explicitly by the physical constants of the plate in the first case, while this is no longer possible in the second case as the relation (79) is no longer valid. In this case, we propose an approximation of the control function with an error that tends to zero. By means of a powerful and systematic approach based on spectral analysis, we improve some already existing results on the optimal rate of the exponential decay and on the analyticity of the associated semigroup.

Citation: Moncef Aouadi, Taoufik Moulahi. The controllability of a thermoelastic plate problem revisited. Evolution Equations & Control Theory, 2018, 7 (1) : 1-31. doi: 10.3934/eect.2018001
References:
[1]

M. Aouadi and T. Moulahi, Approximate controllability of abstract nonsimple thermoelastic problem, J. Evol. Equat. Cont. Theory, 4 (2015), 373-389.  doi: 10.3934/eect.2015.4.373.  Google Scholar

[2]

M. Aouadi and T. Moulahi, Asymptotic analysis of a nonsimple thermoelastic rod, J. Disc. Cont. Dyn. Syst. Serie S, 9 (2016), 1475-1492.  doi: 10.3934/dcdss.2016059.  Google Scholar

[3]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, 2nd ed., Monographs in Mathematics, 96 Birkhäuser, 2011.  Google Scholar

[4]

G. Avalos, Exact controllability of a thermoelastic system with control in the thermal component only, Diff. Int. Equat., 13 (2000), 613-630.   Google Scholar

[5]

G. Avalos and I. Lasiecka, The null controllability of thermoelastic plates and singularity of the associated minimal energy function, J. Math. Anal. Appl., 294 (2004), 34-61.  doi: 10.1016/j.jmaa.2004.01.035.  Google Scholar

[6]

G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system 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.  Google Scholar

[8]

A. Benabdallah and M. G. Naso, Null controllability of a thermoelastic plate, Abstr. Appl. Anal., 7 (2002), 585-599.  doi: 10.1155/S108533750220408X.  Google Scholar

[9]

A. Carrasco, H. Leiva and J. Uzcátegui, Controllability of Semilinear Evolution Equations, Caracas, Venezuela, ISBN 978-980-261-162-1 (2015). Google Scholar

[10]

C. Castro and L. Teresa, Null controllability of the linear system of thermoelastic plates, J. Math. Anal. Appl., 428 (2015), 772-793.  doi: 10.1016/j.jmaa.2015.03.014.  Google Scholar

[11]

S. K. Chang and R. Triggiani, Spectral Analysis of Thermo-elastic Plates, Optimal Control: Theory, Algorithms and Applications, Kluwer, 1998. Google Scholar

[12]

R. Curtain and A. Pritchard, Infinite Dimensional Linear Systems Theory, Lecture Notes in Control and Information Sciences, 8. Springer-Verlag, Berlin-New York, 1978.  Google Scholar

[13]

R. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, vol. 21 of Texts in Applied Mathematics, Springer, New York, NY, USA, 1995.  Google Scholar

[14]

K. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Graduate Texts in Math. vol. 94, Springer-Verlag, 2000.  Google Scholar

[15]

J. Goldstein, Semigroups of Linear Operators and Applications, Oxford University Press, 1985.  Google Scholar

[16]

S. Hansen and B. Y. Zhang, Boundary controllability of a linear thermoelastic beam, J. Math. Anal. Appl., 210 (1997), 182-205.  doi: 10.1006/jmaa.1997.5437.  Google Scholar

[17]

F. Khodja and A. Benabdallah, Sufficient conditions for uniform stabilization of second order equations by dynamic controllers, Dyn. Cont. Disc. Impul. Syst., 7 (2000), 207-222.   Google Scholar

[18]

D. Ya. Khusainov and M. Pokojovy, Solving the linear 1D thermoelasticity equations with pure delay, Int. J. Math. Math. Sci., 2015 (2015), Art. ID 479267, 11 pp.  Google Scholar

[19]

J. Kim, On the energy decay of a linear thermoelastic bar and plate, SIAM J. Math. Anal., 23 (1992), 889-899.  doi: 10.1137/0523047.  Google Scholar

[20]

J. Lagnese, Boundary Stabilization of Thin Plates, SIAM Studies in Applied Mathematics, 10. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989.  Google Scholar

[21]

I. Lasiecka and R. Triggiani, Analyticity, and lack thereof, of thermo-elastic semigroups, ESAIM: Proceedings, Control and partial differential equations, 4 (1998), 199-222.   Google Scholar

[22]

I. Lasiecka and R. Triggiani, Exact null controllability of structurally damped and thermoelastic parabolic models, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl, 9 (1998), 43-69.   Google Scholar

[23]

I. Lasiecka and T. Seidman, Blowup estimates for observability of a thermoelastic system, Asymptot. Anal., 50 (2006), 93-120.   Google Scholar

[24]

G. Lebeau and L. Robbiano, Exact control of the heat equation, Comm. Part, Diff. Eqns., 20 (1995), 335-356.  doi: 10.1080/03605309508821097.  Google Scholar

[25]

G. Lebeau and E. Zuazua, Null controllability of a system of linear thermoelasticity, Arch. Rational Mech. Anal., 141 (1998), 297-329.  doi: 10.1007/s002050050078.  Google Scholar

[26]

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

[27]

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

[28]

H. Leiva and Z. Sivoli, Existence, stability and smoothness of a bounded solution for nonlinear time-varying thermoelastic plate equations, J. Math. Anal. Appl., 285 (2003), 191-211.  doi: 10.1016/S0022-247X(03)00401-3.  Google Scholar

[29]

A. LópezX. Zhang and E. Zuazua, Null controllability of the heat equation as singular limit of the exact controllability of dissipative wave equations, J. Math. Pures Appl., 79 (2000), 741-808.  doi: 10.1016/S0021-7824(99)00144-0.  Google Scholar

[30]

K. Liu and Z. Liu, Exponential stability and analyticity of abstract linear thermoelastic systems, Z. Angew. Math. Phys., 48 (1997), 885-904.  doi: 10.1007/s000330050071.  Google Scholar

[31]

Z. Liu and M. Renardy, A note on the equation of a thermoelastic plate, Appl. Math. Lett., 8 (1995), 1-6.  doi: 10.1016/0893-9659(95)00020-Q.  Google Scholar

[32]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.  Google Scholar

[33]

W. Press, S. Teukolsky, W. Vetterling and B. Flannery, Numerical Recipes in FORTRAN Cambridge University Press, Cambridge, 1992.  Google Scholar

[34]

J. Rivera and L. Fatori, Regularizing properties and propagations of singularities for thermoelastic plates, Math. Meth. Appl. Sci., 21 (1998), 797-821.  doi: 10.1002/(SICI)1099-1476(199806)21:9<797::AID-MMA970>3.0.CO;2-D.  Google Scholar

[35]

J. Rivera and Y. Shibata, A linear thermoelastic plate equation with Dirichlet boundary conditions, Math. Meth. Appl. Sci., 20 (1997), 915-932.  doi: 10.1002/(SICI)1099-1476(19970725)20:11<915::AID-MMA891>3.0.CO;2-4.  Google Scholar

[36]

D. Russell, A general framework for the study of indirect damping mechanisms in elastic systems, J. Math. Anal. Appl., 173 (1993), 339-358.  doi: 10.1006/jmaa.1993.1071.  Google Scholar

[37]

Y. Shibata, On the exponential decay of the energy of a linear thermoelastic plate, Comp. Appl. Math., 13 (1994), 81-102.   Google Scholar

[38]

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

[39]

J. Zabczyk, Mathematical Control Theory: An Introduction, Reprint of the 1995 edition.  Google Scholar

[40]

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 and T. Moulahi, Approximate controllability of abstract nonsimple thermoelastic problem, J. Evol. Equat. Cont. Theory, 4 (2015), 373-389.  doi: 10.3934/eect.2015.4.373.  Google Scholar

[2]

M. Aouadi and T. Moulahi, Asymptotic analysis of a nonsimple thermoelastic rod, J. Disc. Cont. Dyn. Syst. Serie S, 9 (2016), 1475-1492.  doi: 10.3934/dcdss.2016059.  Google Scholar

[3]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, 2nd ed., Monographs in Mathematics, 96 Birkhäuser, 2011.  Google Scholar

[4]

G. Avalos, Exact controllability of a thermoelastic system with control in the thermal component only, Diff. Int. Equat., 13 (2000), 613-630.   Google Scholar

[5]

G. Avalos and I. Lasiecka, The null controllability of thermoelastic plates and singularity of the associated minimal energy function, J. Math. Anal. Appl., 294 (2004), 34-61.  doi: 10.1016/j.jmaa.2004.01.035.  Google Scholar

[6]

G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system 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.  Google Scholar

[8]

A. Benabdallah and M. G. Naso, Null controllability of a thermoelastic plate, Abstr. Appl. Anal., 7 (2002), 585-599.  doi: 10.1155/S108533750220408X.  Google Scholar

[9]

A. Carrasco, H. Leiva and J. Uzcátegui, Controllability of Semilinear Evolution Equations, Caracas, Venezuela, ISBN 978-980-261-162-1 (2015). Google Scholar

[10]

C. Castro and L. Teresa, Null controllability of the linear system of thermoelastic plates, J. Math. Anal. Appl., 428 (2015), 772-793.  doi: 10.1016/j.jmaa.2015.03.014.  Google Scholar

[11]

S. K. Chang and R. Triggiani, Spectral Analysis of Thermo-elastic Plates, Optimal Control: Theory, Algorithms and Applications, Kluwer, 1998. Google Scholar

[12]

R. Curtain and A. Pritchard, Infinite Dimensional Linear Systems Theory, Lecture Notes in Control and Information Sciences, 8. Springer-Verlag, Berlin-New York, 1978.  Google Scholar

[13]

R. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, vol. 21 of Texts in Applied Mathematics, Springer, New York, NY, USA, 1995.  Google Scholar

[14]

K. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Graduate Texts in Math. vol. 94, Springer-Verlag, 2000.  Google Scholar

[15]

J. Goldstein, Semigroups of Linear Operators and Applications, Oxford University Press, 1985.  Google Scholar

[16]

S. Hansen and B. Y. Zhang, Boundary controllability of a linear thermoelastic beam, J. Math. Anal. Appl., 210 (1997), 182-205.  doi: 10.1006/jmaa.1997.5437.  Google Scholar

[17]

F. Khodja and A. Benabdallah, Sufficient conditions for uniform stabilization of second order equations by dynamic controllers, Dyn. Cont. Disc. Impul. Syst., 7 (2000), 207-222.   Google Scholar

[18]

D. Ya. Khusainov and M. Pokojovy, Solving the linear 1D thermoelasticity equations with pure delay, Int. J. Math. Math. Sci., 2015 (2015), Art. ID 479267, 11 pp.  Google Scholar

[19]

J. Kim, On the energy decay of a linear thermoelastic bar and plate, SIAM J. Math. Anal., 23 (1992), 889-899.  doi: 10.1137/0523047.  Google Scholar

[20]

J. Lagnese, Boundary Stabilization of Thin Plates, SIAM Studies in Applied Mathematics, 10. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989.  Google Scholar

[21]

I. Lasiecka and R. Triggiani, Analyticity, and lack thereof, of thermo-elastic semigroups, ESAIM: Proceedings, Control and partial differential equations, 4 (1998), 199-222.   Google Scholar

[22]

I. Lasiecka and R. Triggiani, Exact null controllability of structurally damped and thermoelastic parabolic models, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl, 9 (1998), 43-69.   Google Scholar

[23]

I. Lasiecka and T. Seidman, Blowup estimates for observability of a thermoelastic system, Asymptot. Anal., 50 (2006), 93-120.   Google Scholar

[24]

G. Lebeau and L. Robbiano, Exact control of the heat equation, Comm. Part, Diff. Eqns., 20 (1995), 335-356.  doi: 10.1080/03605309508821097.  Google Scholar

[25]

G. Lebeau and E. Zuazua, Null controllability of a system of linear thermoelasticity, Arch. Rational Mech. Anal., 141 (1998), 297-329.  doi: 10.1007/s002050050078.  Google Scholar

[26]

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

[27]

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

[28]

H. Leiva and Z. Sivoli, Existence, stability and smoothness of a bounded solution for nonlinear time-varying thermoelastic plate equations, J. Math. Anal. Appl., 285 (2003), 191-211.  doi: 10.1016/S0022-247X(03)00401-3.  Google Scholar

[29]

A. LópezX. Zhang and E. Zuazua, Null controllability of the heat equation as singular limit of the exact controllability of dissipative wave equations, J. Math. Pures Appl., 79 (2000), 741-808.  doi: 10.1016/S0021-7824(99)00144-0.  Google Scholar

[30]

K. Liu and Z. Liu, Exponential stability and analyticity of abstract linear thermoelastic systems, Z. Angew. Math. Phys., 48 (1997), 885-904.  doi: 10.1007/s000330050071.  Google Scholar

[31]

Z. Liu and M. Renardy, A note on the equation of a thermoelastic plate, Appl. Math. Lett., 8 (1995), 1-6.  doi: 10.1016/0893-9659(95)00020-Q.  Google Scholar

[32]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.  Google Scholar

[33]

W. Press, S. Teukolsky, W. Vetterling and B. Flannery, Numerical Recipes in FORTRAN Cambridge University Press, Cambridge, 1992.  Google Scholar

[34]

J. Rivera and L. Fatori, Regularizing properties and propagations of singularities for thermoelastic plates, Math. Meth. Appl. Sci., 21 (1998), 797-821.  doi: 10.1002/(SICI)1099-1476(199806)21:9<797::AID-MMA970>3.0.CO;2-D.  Google Scholar

[35]

J. Rivera and Y. Shibata, A linear thermoelastic plate equation with Dirichlet boundary conditions, Math. Meth. Appl. Sci., 20 (1997), 915-932.  doi: 10.1002/(SICI)1099-1476(19970725)20:11<915::AID-MMA891>3.0.CO;2-4.  Google Scholar

[36]

D. Russell, A general framework for the study of indirect damping mechanisms in elastic systems, J. Math. Anal. Appl., 173 (1993), 339-358.  doi: 10.1006/jmaa.1993.1071.  Google Scholar

[37]

Y. Shibata, On the exponential decay of the energy of a linear thermoelastic plate, Comp. Appl. Math., 13 (1994), 81-102.   Google Scholar

[38]

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

[39]

J. Zabczyk, Mathematical Control Theory: An Introduction, Reprint of the 1995 edition.  Google Scholar

[40]

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

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