The exponential stabilization of the higher-dimensional linear system of thermoviscoelasticity

Abstract Using multipler techniques and Lyapunov methods, we prove that the energy in the higher-dimensional linear thermoviscoelasticity decays to zero exponentially by introducing a velocity feedback on part of the boundary of a thermoviscoelastic body, which is clamped along the rest of its boundary, to increase the loss of energy.

[1]  Jacques-Louis Lions,et al.  Modelling Analysis and Control of Thin Plates , 1988 .

[2]  J. Lagnese,et al.  Boundary Stabilization of Linear Elastodynamic Systems , 1983 .

[3]  Fatiha Alabau,et al.  Boundary observability, controllability and stabilization of linear elastodynamic systems , 1999 .

[4]  J. Lagnese.  Boundary Stabilization of Thin Plates , 1987 .

[5]  Sur la décroissance non uniforme de l'énergie dans le système de la thermoélasticité linéaire , 1997 .

[6]  Constantine M. Dafermos,et al.  An abstract Volterra equation with applications to linear viscoelasticity , 1970 .

[7]  W. Desch,et al.  Exponential stabilization of Volterra integrodifferential equations in Hilbert space , 1987 .

[8]  Jaime E. Muñoz Rivera,et al.  A global existence theorem for the Dirichlet problem in nonlinear $n$-dimensional viscoelasticity , 1996, Differential and Integral Equations.

[9]  On boundary feedback stabilization of a viscoelastic membrane , 1989 .

[10]  J. M. Rivera,et al.  Uniform rates of decay in nonlinear viscoelasticity for polynomial decaying kernels , 1996 .

[11]  J. M. Muñoz Rivera,et al.  Decay rates of solutions of an anisotropic inhomogeneousn-dimensional viscoelastic equation with polynomially decaying kernels , 1996 .

[12]  W. A. Day.  The decay of the energy in a viscoelastic body. , 1980 .

[13]  G. Leugering.  A decomposition method for integro-partial differential equations and applications , 1992 .

[14]  T. K. Caughey,et al.  Dynamical Systems and Evolution Equations: Theory and Applications , 1980 .

[15]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .

[16]  D. Mayne.  Nonlinear and Adaptive Control Design [Book Review] , 1996, IEEE Transactions on Automatic Control.

[17]  J. Lagnese.  Decay of solutions of wave equations in a bounded region with boundary dissipation , 1983 .

[18]  E. Zuazua,et al.  A direct method for boundary stabilization of the wave equation , 1990 .

[19]  J. A. Walker,et al.  Dynamical Systems and Evolution Equations , 1980 .

[20]  J. U. Kim.  On the energy decay of a linear thermoelastic bar and plate , 1992 .

[21]  Weijiu Liu,et al.  Partial exact controllability and exponential stability in higher-dimensional linear thermoelasticity , 1998 .

[22]  C. Dafermos.  Asymptotic stability in viscoelasticity , 1970 .

[23]  D. L. Russell.  Review: J.-L. Lions, Controlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués , 1990 .

[24]  Zhuangyi Liu,et al.  A note on the equations of a thermoelastic plate , 1995 .

[25]  Amnon Pazy,et al.  Semigroups of Linear Operators and Applications to Partial Differential Equations , 1992, Applied Mathematical Sciences.

[26]  E. Zuazua,et al.  Uniform stabilization of the wave equation by nonlinear boundary feedback , 1990 .

[27]  P. Grisvard.  Contrôlabilité exacte des solutions de l'équation des ondes en présence de singularités , 1989 .

[28]  Zhuangyi Liu,et al.  On the exponential stability of linear viscoelasticity and thermoviscoelasticity , 1996 .

[29]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[30]  Zhuangyi Liu,et al.  Exponential stability of the semigroup associated with a thermoelastic system , 1993 .

[31]  C. B. Navarro.  Asymptotic stability in linear thermoviscoelasticity , 1978 .

[32]  C. DeWitt-Morette,et al.  Mathematical Analysis and Numerical Methods for Science and Technology , 1990 .

[33]  Scott W. Hansen,et al.  Exponential energy decay in a linear thermoelastic rod , 1992 .

[34]  Global smooth solutions for the Cauchy problem in nonlinear viscoelasticity , 1994 .

[35]  Constantine M. Dafermos,et al.  On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity , 1968 .