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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 & Control Theory, 2016, 5 (2) : 201-224. doi: 10.3934/eect.2016001
References:
[1]

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

[2]

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

[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.  Google Scholar

[4]

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

[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. Google Scholar

[6]

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

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[30]

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

[31]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

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

show all references

References:
[1]

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

[2]

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

[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.  Google Scholar

[4]

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

[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. Google Scholar

[6]

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

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[30]

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

[31]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

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

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