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

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

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.
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, (1975).   Google Scholar

[2]

S. Agmon, Lectures on Elliptic Boundary Value Problems,, D. Van Nostrand Company, (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.  doi: 10.1137/S0363012996313835.  Google Scholar

[4]

M. Aouadi, Generalized theory of thermoelastic diffusion for anisotropic media,, J. Therm. Stresses, 31 (2008), 270.  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.   Google Scholar

[6]

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

[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.  Google Scholar

[8]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces,, Noordhoff, (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.  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.  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.  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.  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.   Google Scholar

[14]

S. W. Hansen, Boundary control of a one-dimentional linear thermoelastic rod,, SIAM J. Control Optim., 32 (1994), 1052.  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.   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.   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.   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.   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, 69 (2008), 898.  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.  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.  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.  doi: 10.1007/s002050050078.  Google Scholar

[24]

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

[29]

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

[30]

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

[31]

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

[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.  Google Scholar

[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.  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, 10 (1974), 291.   Google Scholar

[35]

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

show all references

References:
[1]

R. Adams, Sobolev Spaces,, Academic Press, (1975).   Google Scholar

[2]

S. Agmon, Lectures on Elliptic Boundary Value Problems,, D. Van Nostrand Company, (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.  doi: 10.1137/S0363012996313835.  Google Scholar

[4]

M. Aouadi, Generalized theory of thermoelastic diffusion for anisotropic media,, J. Therm. Stresses, 31 (2008), 270.  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.   Google Scholar

[6]

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

[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.  Google Scholar

[8]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces,, Noordhoff, (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.  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.  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.  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.  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.   Google Scholar

[14]

S. W. Hansen, Boundary control of a one-dimentional linear thermoelastic rod,, SIAM J. Control Optim., 32 (1994), 1052.  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.   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.   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.   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.   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, 69 (2008), 898.  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.  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.  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.  doi: 10.1007/s002050050078.  Google Scholar

[24]

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

[29]

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

[30]

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

[31]

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

[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.  Google Scholar

[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.  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, 10 (1974), 291.   Google Scholar

[35]

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

[1]

Boris Andreianov, Mohamed Maliki. On classes of well-posedness for quasilinear diffusion equations in the whole space. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 505-531. doi: 10.3934/dcdss.2020361

[2]

Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302

[3]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382

[4]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[5]

Tong Tang, Jianzhu Sun. Local well-posedness for the density-dependent incompressible magneto-micropolar system with vacuum. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020377

[6]

Xiaopeng Zhao, Yong Zhou. Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 795-813. doi: 10.3934/dcdsb.2020142

[7]

Dongfen Bian, Yao Xiao. Global well-posedness of non-isothermal inhomogeneous nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1243-1272. doi: 10.3934/dcdsb.2020161

[8]

Mokhtari Yacine. Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021004

[9]

Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434

[10]

Michael Winkler, Christian Stinner. Refined regularity and stabilization properties in a degenerate haptotaxis system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 4039-4058. doi: 10.3934/dcds.2020030

[11]

Hai-Yang Jin, Zhi-An Wang. Global stabilization of the full attraction-repulsion Keller-Segel system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3509-3527. doi: 10.3934/dcds.2020027

[12]

Jean-Paul Chehab. Damping, stabilization, and numerical filtering for the modeling and the simulation of time dependent PDEs. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021002

[13]

Yubiao Liu, Chunguo Zhang, Tehuan Chen. Stabilization of 2-d Mindlin-Timoshenko plates with localized acoustic boundary feedback. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021006

[14]

Guangjun Shen, Xueying Wu, Xiuwei Yin. Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 755-774. doi: 10.3934/dcdsb.2020133

[15]

Xiaorui Wang, Genqi Xu, Hao Chen. Uniform stabilization of 1-D Schrödinger equation with internal difference-type control. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021022

[16]

Peizhao Yu, Guoshan Zhang, Yi Zhang. Decoupling of cubic polynomial matrix systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 13-26. doi: 10.3934/naco.2020012

[17]

Xin Guo, Lexin Li, Qiang Wu. Modeling interactive components by coordinate kernel polynomial models. Mathematical Foundations of Computing, 2020, 3 (4) : 263-277. doi: 10.3934/mfc.2020010

[18]

Jan Bouwe van den Berg, Elena Queirolo. A general framework for validated continuation of periodic orbits in systems of polynomial ODEs. Journal of Computational Dynamics, 2021, 8 (1) : 59-97. doi: 10.3934/jcd.2021004

[19]

Nguyen Thi Kim Son, Nguyen Phuong Dong, Le Hoang Son, Alireza Khastan, Hoang Viet Long. Complete controllability for a class of fractional evolution equations with uncertainty. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020104

[20]

Oleg Yu. Imanuvilov, Jean Pierre Puel. On global controllability of 2-D Burgers equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 299-313. doi: 10.3934/dcds.2009.23.299

2019 Impact Factor: 0.953

Metrics

  • PDF downloads (46)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]