On the exponential stability of linear viscoelasticity and thermoviscoelasticity
Authors:
Zhuangyi Liu and Songmu Zheng
Journal:
Quart. Appl. Math. 54 (1996), 21-31
MSC:
Primary 73F15; Secondary 35B35, 35Q72, 73B30
DOI:
https://doi.org/10.1090/qam/1373836
MathSciNet review:
MR1373836
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: The exponential stability of the semigroup associated with one-dimensional linear viscoelastic and thermoviscoelastic equations with several types of boundary conditions is proved for a class of kernel functions, including the weakly singular kernels.
J. A. Burns and R. H. Fabiano, Feedback control of a hyperbolic partial differential equation with viscoelastic damping, Control Theory and Advanced Technology, Vol. 5, No. 2, 1989, pp. 157–188
J. A. Burns, Z. Liu, and S. Zheng, On the energy decay of a linear thermoelastic bar, J. Math. Anal. Appl. (to appear)
W. A. Day, The decay of energy in a viscoelastic body, Mathematika, Vol. 27, 1980, pp. 268–286
C. M. Dafermos, On the existence and the asymptotic stability of solution to the equations of linear thermoelasticity, Arch. Rat. Mech. Anal. 29, 241–271 (1968)
C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rat. Mech. Anal. 37, 297–308 (1970)
C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity, J. Differential Equations 7, 554–569 (1970)
W. Desch and R. K. Miller, Exponential stabilization of Volterra integrodifferential equations in Hilbert space, J. Differential Equations 70, 366–389 (1987)
W. Desch and R. K. Miller, Exponential stabilization of Volterra integral equations with singular kernels, J. Integral Equations Appl. 1, 397–433 (1988)
R. H. Fabiano and K. Ito, Semigroup theory and numerical approximation for equations in linear viscoelasticity, SIAM J. Math. Anal. 21 (2), 374–393 (1990)
R. H. Fabiano and K. Ito, An approximation framework for equations in linear viscoelasticity with strongly singular kernels, Quart. Appl. Math. 52, 65–81 (1994)
M. Fabrizio and B. Lazzari, On the existence and asymptotic stability of solutions for linearly viscoelastic solids, Arch. Rat. Mech. Anal. 116, 139–152 (1991)
J. S. Gibson, I. G. Rosen, and G. Tao, Approximation in control of thermoelastic systems, SIAM J. Control and Optimization 30 5, 1163–1189 (1992)
S. W. Hansen, Exponential energy decay in a linear thermoelastic rod, J. Math. Anal. Appl. 167, 429–442 (1992)
F. L. Huang, Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations 1 (1), 43–56 (1985)
K. B. Hannsgen, Y. Renardy, and R. L. Wheeler, Effectiveness and robustness with respect to time delays of boundary feedback stabilization in one-dimensional viscoelasticity, SIAM J. Control Optim. 26 5, 1200–1233 (1988)
K. B. Hannsgen and R. L. Wheeler, Viscoelastic and boundary feedback damping: Precise energy decay rates when creep modes are dominant, J. Integral Equations Appl. 2, 495–527 (1990)
K. B. Hannsgen and R. L. Wheeler, Moment conditions for a Volterra integral equation in a Banach space, Proceedings of Delay Differential Equations and Dynamical Systems, Claremont, Vol. 1475 of Springer Lecture Notes in Mathematics, Springer-Verlag, 1991, pp. 204–209
S. Jiang, Global solution of the Neumann problem in one-dimensional nonlinear thermoelasticity, preprint, 1991
J. U. Kim, On the energy decay of a linear thermoelastic bar and plate, SIAM J. Math. Anal. 23 (4), 889–899 (1992)
Z. Liu, Approximation and control of a thermoviscoelastic system, Ph.D. dissertation, Virginia Polytechnic Institute and State University, Department of Mathematics, August, 1989
J. Lagnese, Boundary Stabilization of Thin Plates, Vol. 10 of SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, 1989
G. Leugering, On boundary feedback stabilization of a viscoelastic membrane, Dynamics and Stability of Systems 4 (1), 71–79 (1989)
G. Leugering, On boundary feedback stabilizability of a viscoelastic beam, Proc. Roy. Soc. Edinburgh Sect. A 114 (1), 57–69 (1990)
Z. Liu and S. Zheng, Exponential stability of the semigroup associated with a thermoelastic system, Quart. Appl. Math. 51, 535–545 (1993)
C. B. Navaro, Asymptotic stability in linear thermoviscoelasticity, J. Math. Appl. 65, 399–431 (1978)
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential equations, Springer, New York, 1983
J. E. M. Rivera, Energy decay rate in linear thermoelasticity, Funkcial Ekvac. 35, 19–30 (1992)
M. Slemrod, Global existence, uniqueness, and asymptotic stability of classical smooth solutions in one-dimensional nonlinear thermoelasticity, Arch Rat. Mech. Anal. 76, 97–133 (1981)
Y. Shibata, Neumann problem for one-dimensional nonlinear thermoelasticity, preprint, 1991
J. A. Burns and R. H. Fabiano, Feedback control of a hyperbolic partial differential equation with viscoelastic damping, Control Theory and Advanced Technology, Vol. 5, No. 2, 1989, pp. 157–188
J. A. Burns, Z. Liu, and S. Zheng, On the energy decay of a linear thermoelastic bar, J. Math. Anal. Appl. (to appear)
W. A. Day, The decay of energy in a viscoelastic body, Mathematika, Vol. 27, 1980, pp. 268–286
C. M. Dafermos, On the existence and the asymptotic stability of solution to the equations of linear thermoelasticity, Arch. Rat. Mech. Anal. 29, 241–271 (1968)
C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rat. Mech. Anal. 37, 297–308 (1970)
C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity, J. Differential Equations 7, 554–569 (1970)
W. Desch and R. K. Miller, Exponential stabilization of Volterra integrodifferential equations in Hilbert space, J. Differential Equations 70, 366–389 (1987)
W. Desch and R. K. Miller, Exponential stabilization of Volterra integral equations with singular kernels, J. Integral Equations Appl. 1, 397–433 (1988)
R. H. Fabiano and K. Ito, Semigroup theory and numerical approximation for equations in linear viscoelasticity, SIAM J. Math. Anal. 21 (2), 374–393 (1990)
R. H. Fabiano and K. Ito, An approximation framework for equations in linear viscoelasticity with strongly singular kernels, Quart. Appl. Math. 52, 65–81 (1994)
M. Fabrizio and B. Lazzari, On the existence and asymptotic stability of solutions for linearly viscoelastic solids, Arch. Rat. Mech. Anal. 116, 139–152 (1991)
J. S. Gibson, I. G. Rosen, and G. Tao, Approximation in control of thermoelastic systems, SIAM J. Control and Optimization 30 5, 1163–1189 (1992)
S. W. Hansen, Exponential energy decay in a linear thermoelastic rod, J. Math. Anal. Appl. 167, 429–442 (1992)
F. L. Huang, Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations 1 (1), 43–56 (1985)
K. B. Hannsgen, Y. Renardy, and R. L. Wheeler, Effectiveness and robustness with respect to time delays of boundary feedback stabilization in one-dimensional viscoelasticity, SIAM J. Control Optim. 26 5, 1200–1233 (1988)
K. B. Hannsgen and R. L. Wheeler, Viscoelastic and boundary feedback damping: Precise energy decay rates when creep modes are dominant, J. Integral Equations Appl. 2, 495–527 (1990)
K. B. Hannsgen and R. L. Wheeler, Moment conditions for a Volterra integral equation in a Banach space, Proceedings of Delay Differential Equations and Dynamical Systems, Claremont, Vol. 1475 of Springer Lecture Notes in Mathematics, Springer-Verlag, 1991, pp. 204–209
S. Jiang, Global solution of the Neumann problem in one-dimensional nonlinear thermoelasticity, preprint, 1991
J. U. Kim, On the energy decay of a linear thermoelastic bar and plate, SIAM J. Math. Anal. 23 (4), 889–899 (1992)
Z. Liu, Approximation and control of a thermoviscoelastic system, Ph.D. dissertation, Virginia Polytechnic Institute and State University, Department of Mathematics, August, 1989
J. Lagnese, Boundary Stabilization of Thin Plates, Vol. 10 of SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, 1989
G. Leugering, On boundary feedback stabilization of a viscoelastic membrane, Dynamics and Stability of Systems 4 (1), 71–79 (1989)
G. Leugering, On boundary feedback stabilizability of a viscoelastic beam, Proc. Roy. Soc. Edinburgh Sect. A 114 (1), 57–69 (1990)
Z. Liu and S. Zheng, Exponential stability of the semigroup associated with a thermoelastic system, Quart. Appl. Math. 51, 535–545 (1993)
C. B. Navaro, Asymptotic stability in linear thermoviscoelasticity, J. Math. Appl. 65, 399–431 (1978)
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential equations, Springer, New York, 1983
J. E. M. Rivera, Energy decay rate in linear thermoelasticity, Funkcial Ekvac. 35, 19–30 (1992)
M. Slemrod, Global existence, uniqueness, and asymptotic stability of classical smooth solutions in one-dimensional nonlinear thermoelasticity, Arch Rat. Mech. Anal. 76, 97–133 (1981)
Y. Shibata, Neumann problem for one-dimensional nonlinear thermoelasticity, preprint, 1991
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
73F15,
35B35,
35Q72,
73B30
Retrieve articles in all journals
with MSC:
73F15,
35B35,
35Q72,
73B30
Additional Information
Article copyright:
© Copyright 1996
American Mathematical Society
