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Partial exact controllability for inhomogeneous multidimensional thermoelastic diffusion problem
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 |
References:
[1] |
R. Adams, Sobolev Spaces,, Academic Press, (1975).
|
[2] |
S. Agmon, Lectures on Elliptic Boundary Value Problems,, D. Van Nostrand Company, (1965).
|
[3] |
F. Alabau and V. Komornik, Boundary observability, controllability and stabilization of linear elastodynamic systems,, SIAM J. Control Optim., 37 (1999), 521.
doi: 10.1137/S0363012996313835. |
[4] |
M. Aouadi, Generalized theory of thermoelastic diffusion for anisotropic media,, J. Therm. Stresses, 31 (2008), 270.
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. Google Scholar |
[6] |
R. F. Apolaya, Exact controllability for temporally wave equation,, Portugaliae Mathematica, 51 (1994), 475.
|
[7] |
M. Assila, Nonlinear boundary stabilization of an inhomogeneous and anisotropic thermoelasticity system,, App. Math. Lett., 13 (2000), 71.
doi: 10.1016/S0893-9659(99)00147-0. |
[8] |
V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces,, Noordhoff, (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.
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.
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.
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.
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.
|
[14] |
S. W. Hansen, Boundary control of a one-dimentional linear thermoelastic rod,, SIAM J. Control Optim., 32 (1994), 1052.
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.
|
[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.
|
[17] |
V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation,, J. Math. Pures Appl., 69 (1990), 33.
|
[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.
|
[20] |
I. Lasiecka and D. Toundykov, Regularity of higher energies of wave equation with nonlinear localized damping and a nonlinear source,, Nonlinear Analysis: Theory, 69 (2008), 898.
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.
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.
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.
doi: 10.1007/s002050050078. |
[24] |
J. L. Lions, Contôlabilté Exacte Perturbations et Stabilisations de Systèmes Distribués, Tome 2., Pertubations, (1988).
|
[25] |
W. J. Liu, Partial exact controllability and exponential stability in higher-dimensional linear thermoelasticity,, ESAIM: Control Optim. Calc. Var., 3 (1998), 23.
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.
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.
|
[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.
|
[29] |
J. E. Munoz Rivera and M. L. Olivera, Stability in inhomogeneous and anisotropic thermoelasticity,, Bollettino U.M.I, 11 (1997), 115.
|
[30] |
A. K. Nandakumaran and R. K. George, Partial exact controllability of linear thermoelastic system,, Indian J. Math, 37 (1995), 165.
|
[31] |
K. Narukawa, Boundary value control of thermoelastic systems,, Hiroshima Math. J., 13 (1983), 227.
|
[32] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences, (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.
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, 10 (1974), 291.
|
[35] |
E. Zuazua, Controllability of the linear system of thermoelasticity,, J. Math. Pures Appl., 74 (1995), 303.
|
show all references
References:
[1] |
R. Adams, Sobolev Spaces,, Academic Press, (1975).
|
[2] |
S. Agmon, Lectures on Elliptic Boundary Value Problems,, D. Van Nostrand Company, (1965).
|
[3] |
F. Alabau and V. Komornik, Boundary observability, controllability and stabilization of linear elastodynamic systems,, SIAM J. Control Optim., 37 (1999), 521.
doi: 10.1137/S0363012996313835. |
[4] |
M. Aouadi, Generalized theory of thermoelastic diffusion for anisotropic media,, J. Therm. Stresses, 31 (2008), 270.
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. Google Scholar |
[6] |
R. F. Apolaya, Exact controllability for temporally wave equation,, Portugaliae Mathematica, 51 (1994), 475.
|
[7] |
M. Assila, Nonlinear boundary stabilization of an inhomogeneous and anisotropic thermoelasticity system,, App. Math. Lett., 13 (2000), 71.
doi: 10.1016/S0893-9659(99)00147-0. |
[8] |
V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces,, Noordhoff, (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.
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.
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.
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.
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.
|
[14] |
S. W. Hansen, Boundary control of a one-dimentional linear thermoelastic rod,, SIAM J. Control Optim., 32 (1994), 1052.
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.
|
[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.
|
[17] |
V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation,, J. Math. Pures Appl., 69 (1990), 33.
|
[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.
|
[20] |
I. Lasiecka and D. Toundykov, Regularity of higher energies of wave equation with nonlinear localized damping and a nonlinear source,, Nonlinear Analysis: Theory, 69 (2008), 898.
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.
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.
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.
doi: 10.1007/s002050050078. |
[24] |
J. L. Lions, Contôlabilté Exacte Perturbations et Stabilisations de Systèmes Distribués, Tome 2., Pertubations, (1988).
|
[25] |
W. J. Liu, Partial exact controllability and exponential stability in higher-dimensional linear thermoelasticity,, ESAIM: Control Optim. Calc. Var., 3 (1998), 23.
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.
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.
|
[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.
|
[29] |
J. E. Munoz Rivera and M. L. Olivera, Stability in inhomogeneous and anisotropic thermoelasticity,, Bollettino U.M.I, 11 (1997), 115.
|
[30] |
A. K. Nandakumaran and R. K. George, Partial exact controllability of linear thermoelastic system,, Indian J. Math, 37 (1995), 165.
|
[31] |
K. Narukawa, Boundary value control of thermoelastic systems,, Hiroshima Math. J., 13 (1983), 227.
|
[32] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences, (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.
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, 10 (1974), 291.
|
[35] |
E. Zuazua, Controllability of the linear system of thermoelasticity,, J. Math. Pures Appl., 74 (1995), 303.
|
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