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June  2016, 5(2): 201-224. doi: 10.3934/eect.2016001

Partial exact controllability for inhomogeneous multidimensional thermoelastic diffusion problem

Moncef Aouadi 1, and  Kaouther Boulehmi 2,

1. 

Ecole Nationale d'Ingénieurs de Bizerte, Université de Carthage, BP66, Campus Universitaire Menzel Abderrahman 7035
2. 

Faculté des Sciences de Bizerte, Jarzouna 7021, Université de Carthage, Tunisia

Received  January 2016 Revised  May 2016 Published  June 2016

The problem of stabilization and controllability for inhomogeneous multidimensional thermoelastic diffusion problem is considered for anisotropic material. By introducing a nonlinear feedback function on part of the boundary of the material, which is clamped along the rest of its boundary, we prove that the energy of the system decays to zero exponentially or polynomially. Both rates of decay are determined explicitly by the physical parameters. Via Russell's ``Controllability via Stabilizability" principle, we prove that the considered system is partially controllable by a boundary function determined explicitly.
Keywords: Thermoelastic diffusion, exponential stabilization, well-posedness, controllability., polynomial stabilization.
Mathematics Subject Classification: Primary: 35B37; Secondary: 35B3.
Citation: Moncef Aouadi, Kaouther Boulehmi. Partial exact controllability for inhomogeneous multidimensional thermoelastic diffusion problem. Evolution Equations and Control Theory, 2016, 5 (2) : 201-224. doi: 10.3934/eect.2016001
References:
[1]

R. Adams, Sobolev Spaces, Academic Press, New York, 1975.

[2]

S. Agmon, Lectures on Elliptic Boundary Value Problems, D. Van Nostrand Company, Inc., Princeton, 1965.

[3]

F. Alabau and V. Komornik, Boundary observability, controllability and stabilization of linear elastodynamic systems, SIAM J. Control Optim., 37 (1999), 521-542. doi: 10.1137/S0363012996313835.

[4]

M. Aouadi, Generalized theory of thermoelastic diffusion for anisotropic media, J. Therm. Stresses, 31 (2008), 270-285. doi: 10.1080/01495730701876742.

[5]

M. Aouadi and T. Moulahi, Optimal decay rate for unidimensional thermoelastic problem within the Green-Lindsay model, J. Therm. Stresses, 38 (2015), 1199-1216.

[6]

R. F. Apolaya, Exact controllability for temporally wave equation, Portugaliae Mathematica, 51 (1994), 475-488.

[7]

M. Assila, Nonlinear boundary stabilization of an inhomogeneous and anisotropic thermoelasticity system, App. Math. Lett., 13 (2000), 71-76. doi: 10.1016/S0893-9659(99)00147-0.

[8]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, The Netherlands, 1976.

[9]

P. Barral and P. Quintela, A numerical method for simulation of thermal stresses during casting of aluminium slabs, Comput. Methods Appl. Mech. Eng., 178 (1998), 69-88. doi: 10.1016/S0045-7825(99)00005-5.

[10]

A. Bermudez, M. C. Muniz and P. Quintela, Numerical solution of a three-dimensional thermoelectric problem taking place in an aluminium electrolytic cell, Comput. Methods Appl. Mech. Eng., 106 (1993), 129-142. doi: 10.1016/0045-7825(93)90188-4.

[11]

K. Boulehmi and M. Aouadi, Decay of solutions in inhomogeneous thermoelastic diffusion bars, Appl. Anal., 93 (2014), 281-304. doi: 10.1080/00036811.2013.769133.

[12]

C. M. Dafermos, On the existence and the asymptotic stability of solution to the equation of linear thermoelasticity, Arch. Rat. Mech. Anal., 29 (1968), 241-271. doi: 10.1007/BF00276727.

[13]

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

[14]

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

[15]

M. A. Horn, Sharp trace regularity for the solutions of the equations of dynamic elasticity, J. Math. Syst. Estim. Control, 8 (1998), 217-219.

[16]

S. Jian, J. E. Munoz Rivera and R. Racke, Asymptotic stability and global existence in thermoelasticity with symmetry, Quart. Appl. Math., 56 (1998), 259-275.

[17]

V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation, J. Math. Pures Appl., 69 (1990), 33-54.

[18]

I. Lasiecka, Mathematical Control Theory of Coupled PDEs-Lecture Notes, CBMS-NSF Regional Conference Series in Applied Mathematics SIAM, 75 (2002). doi: 10.1137/1.9780898717099.

[19]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Diff. Int. Equat., 6 (1993), 507-533.

[20]

I. Lasiecka and D. Toundykov, Regularity of higher energies of wave equation with nonlinear localized damping and a nonlinear source, Nonlinear Analysis: Theory, Methods and Applications, 69 (2008), 898-910. doi: 10.1016/j.na.2008.02.069.

[21]

I. Lasiecka and R. Triggiani, Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions, Appl. Math. Optim., 25 (1992), 189-224. doi: 10.1007/BF01182480.

[22]

G. Lebeau and E. Zuazua, Sur la décroissance non uniforme de l'énergie dans le système de la thermoélasticité linéaire, C. R. Acad. Sci. Paris Sr. I Math, 324 (1997), 409-415. doi: 10.1016/S0764-4442(97)80077-8.

[23]

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

[24]

J. L. Lions, Contôlabilté Exacte Perturbations et Stabilisations de Systèmes Distribués, Tome 2. Pertubations, Masson, Paris, 1988.

[25]

W. J. Liu, Partial exact controllability and exponential stability in higher-dimensional linear thermoelasticity, ESAIM: Control Optim. Calc. Var., 3 (1998), 23-48. doi: 10.1051/cocv:1998101.

[26]

W. J. Liu, Correction to "Partial exact controllability and exponential stability in higher-dimensional linear thermoelasticity", ESAIM: Control Optim. Calc. Var., 3 (1998), 323-327. doi: 10.1051/cocv:1998113.

[27]

W. J. Liu and G. H. Williams, Partial exact controllability for the linear thermo-viscoelastic model, Electron. J. Differential Equations, 1998 (1998), 1-11.

[28]

W. J. Liu and E. Zuazua, Uniform stabilization of higher-dimensional system of thermoelasticity with a nonlinear boundary feedback, Quart. Appl. Math., 59 (2001), 269-314.

[29]

J. E. Munoz Rivera and M. L. Olivera, Stability in inhomogeneous and anisotropic thermoelasticity, Bollettino U.M.I, 11 (1997), 115-127.

[30]

A. K. Nandakumaran and R. K. George, Partial exact controllability of linear thermoelastic system, Indian J. Math, 37 (1995), 165-174.

[31]

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

[32]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[33]

D. C. Pereira and G. P. Menzala, Exponential stability in linear thermoelasticity: The inhomogeneous case, Appl. Anal., 44 (1992), 21-35. doi: 10.1080/00036819208840066.

[34]

D. L. Russell, Exact boundary value controlability theorems for wave and heat processes in star-complemented regions, in Differential games and control theory, Roxin, Lui, and Sternberg, Eds., Marcel Dekker Inc., New York, 10 (1974), 291-319.

[35]

E. Zuazua, Controllability of the linear system of thermoelasticity, J. Math. Pures Appl., 74 (1995), 303-346.

show all references

References:
[1]

R. Adams, Sobolev Spaces, Academic Press, New York, 1975.

[2]

S. Agmon, Lectures on Elliptic Boundary Value Problems, D. Van Nostrand Company, Inc., Princeton, 1965.

[3]

F. Alabau and V. Komornik, Boundary observability, controllability and stabilization of linear elastodynamic systems, SIAM J. Control Optim., 37 (1999), 521-542. doi: 10.1137/S0363012996313835.

[4]

M. Aouadi, Generalized theory of thermoelastic diffusion for anisotropic media, J. Therm. Stresses, 31 (2008), 270-285. doi: 10.1080/01495730701876742.

[5]

M. Aouadi and T. Moulahi, Optimal decay rate for unidimensional thermoelastic problem within the Green-Lindsay model, J. Therm. Stresses, 38 (2015), 1199-1216.

[6]

R. F. Apolaya, Exact controllability for temporally wave equation, Portugaliae Mathematica, 51 (1994), 475-488.

[7]

M. Assila, Nonlinear boundary stabilization of an inhomogeneous and anisotropic thermoelasticity system, App. Math. Lett., 13 (2000), 71-76. doi: 10.1016/S0893-9659(99)00147-0.

[8]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, The Netherlands, 1976.

[9]

P. Barral and P. Quintela, A numerical method for simulation of thermal stresses during casting of aluminium slabs, Comput. Methods Appl. Mech. Eng., 178 (1998), 69-88. doi: 10.1016/S0045-7825(99)00005-5.

[10]

A. Bermudez, M. C. Muniz and P. Quintela, Numerical solution of a three-dimensional thermoelectric problem taking place in an aluminium electrolytic cell, Comput. Methods Appl. Mech. Eng., 106 (1993), 129-142. doi: 10.1016/0045-7825(93)90188-4.

[11]

K. Boulehmi and M. Aouadi, Decay of solutions in inhomogeneous thermoelastic diffusion bars, Appl. Anal., 93 (2014), 281-304. doi: 10.1080/00036811.2013.769133.

[12]

C. M. Dafermos, On the existence and the asymptotic stability of solution to the equation of linear thermoelasticity, Arch. Rat. Mech. Anal., 29 (1968), 241-271. doi: 10.1007/BF00276727.

[13]

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

[14]

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

[15]

M. A. Horn, Sharp trace regularity for the solutions of the equations of dynamic elasticity, J. Math. Syst. Estim. Control, 8 (1998), 217-219.

[16]

S. Jian, J. E. Munoz Rivera and R. Racke, Asymptotic stability and global existence in thermoelasticity with symmetry, Quart. Appl. Math., 56 (1998), 259-275.

[17]

V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation, J. Math. Pures Appl., 69 (1990), 33-54.

[18]

I. Lasiecka, Mathematical Control Theory of Coupled PDEs-Lecture Notes, CBMS-NSF Regional Conference Series in Applied Mathematics SIAM, 75 (2002). doi: 10.1137/1.9780898717099.

[19]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Diff. Int. Equat., 6 (1993), 507-533.

[20]

I. Lasiecka and D. Toundykov, Regularity of higher energies of wave equation with nonlinear localized damping and a nonlinear source, Nonlinear Analysis: Theory, Methods and Applications, 69 (2008), 898-910. doi: 10.1016/j.na.2008.02.069.

[21]

I. Lasiecka and R. Triggiani, Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions, Appl. Math. Optim., 25 (1992), 189-224. doi: 10.1007/BF01182480.

[22]

G. Lebeau and E. Zuazua, Sur la décroissance non uniforme de l'énergie dans le système de la thermoélasticité linéaire, C. R. Acad. Sci. Paris Sr. I Math, 324 (1997), 409-415. doi: 10.1016/S0764-4442(97)80077-8.

[23]

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

[24]

J. L. Lions, Contôlabilté Exacte Perturbations et Stabilisations de Systèmes Distribués, Tome 2. Pertubations, Masson, Paris, 1988.

[25]

W. J. Liu, Partial exact controllability and exponential stability in higher-dimensional linear thermoelasticity, ESAIM: Control Optim. Calc. Var., 3 (1998), 23-48. doi: 10.1051/cocv:1998101.

[26]

W. J. Liu, Correction to "Partial exact controllability and exponential stability in higher-dimensional linear thermoelasticity", ESAIM: Control Optim. Calc. Var., 3 (1998), 323-327. doi: 10.1051/cocv:1998113.

[27]

W. J. Liu and G. H. Williams, Partial exact controllability for the linear thermo-viscoelastic model, Electron. J. Differential Equations, 1998 (1998), 1-11.

[28]

W. J. Liu and E. Zuazua, Uniform stabilization of higher-dimensional system of thermoelasticity with a nonlinear boundary feedback, Quart. Appl. Math., 59 (2001), 269-314.

[29]

J. E. Munoz Rivera and M. L. Olivera, Stability in inhomogeneous and anisotropic thermoelasticity, Bollettino U.M.I, 11 (1997), 115-127.

[30]

A. K. Nandakumaran and R. K. George, Partial exact controllability of linear thermoelastic system, Indian J. Math, 37 (1995), 165-174.

[31]

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

[32]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[33]

D. C. Pereira and G. P. Menzala, Exponential stability in linear thermoelasticity: The inhomogeneous case, Appl. Anal., 44 (1992), 21-35. doi: 10.1080/00036819208840066.

[34]

D. L. Russell, Exact boundary value controlability theorems for wave and heat processes in star-complemented regions, in Differential games and control theory, Roxin, Lui, and Sternberg, Eds., Marcel Dekker Inc., New York, 10 (1974), 291-319.

[35]

E. Zuazua, Controllability of the linear system of thermoelasticity, J. Math. Pures Appl., 74 (1995), 303-346.

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