An approximation framework for equations in linear viscoelasticity with strongly singular kernels
Authors:
R. H. Fabiano and K. Ito
Journal:
Quart. Appl. Math. 52 (1994), 65-81
MSC:
Primary 34K30; Secondary 45K05, 47D06, 47N20, 65R20, 73F15
DOI:
https://doi.org/10.1090/qam/1262320
MathSciNet review:
MR1262320
Full-text PDF Free Access
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Abstract: In this paper we consider equations that arise in linear viscoelastic models. Within the context of linear semigroup theory we present an approximation framework for these equations. A relevant convergence result is proved using the Trotter-Kato theorem. This work extends previous results that did not apply to equations with strongly singular kernels.
H. T. Banks and J. A. Burns, Hereditary control problems: numerical methods based on averaging approximations, SIAM J. Control Optim. 16, 169–208 (1978)
H. T. Banks, R. H. Fabiano, and Y. Wang, Estimation of Boltzmann damping coefficients in beam models, LCDS/CCS Report 88-13, Brown University, Providence, RI, 1988
H. T. Banks, R. H. Fabiano, Y. Wang, D. J. Inman, and H. Cudney, Spatial versus time hysteresis in damping mechanisms, 27th IEEE Conference on Decision and Control, 1988, pp. 1674–1677
J. A. Burns and R. H. Fabiano, Feedback control of a hyperbolic partial differential equation with viscoelastic damping, Control Theory Adv. Tech. 5, 157–188 (1989)
J. A. Burns, E. M. Cliff, Z. Y. Liu, and R. E. Miller, Control of a thermoviscoelastic system, 27th IEEE Conference on Decision and Control, 1988, pp. 1249–1252
J. A. Burns, Z. Y. Liu, and R. E. Miller, Approximations of thermoelastic and viscoelastic control systems, Numer. Funct. Anal. Optim. 12, 79–135 (1991)
C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal. 37, 297–308 (1979)
C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity, J. Differential Equations 7, 554–569 (1970)
C. M. Dafermos, Contraction semigroups and trend to equilibrium in continuum mechanics, Applications of Methods of Functional Analysis to Problems in Mechanics, Lecture Notes in Math., no. 503, Springer-Verlag, New York, 1976, pp. 295–306
W. Desch and R. K. Miller, Exponential stabilization of Volterra integrodifferential equations in Hilbert spaces, J. Differential Equations 70, 366–389 (1987)
R. H. Fabiano and K. Ito, Semigroup theory and numerical approximation for equations in linear viscoelasticity, SIAM J. Math. Anal. 21, 374–393 (1990)
R. H. Fabiano and K. Ito, Semigroup theory in linear viscoelasticity: Weakly and strongly singular kernels, Internat. Ser. Numer. Math., vol. 91, Birkhäuser, Basel, 1989, pp. 109–121
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, 1200–1234 (1988)
K. B. Hannsgen and R. L. Wheeler, Time delays and boundary feedback stabilization in one-dimensional viscoelasticity, Distributed Parameter Systems, Lecture Notes in Control and Inform. Sci., Springer, New York, 1987, pp. 136–152
T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1976
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983
H. Tanabe, Equations of Evolution, Pitman, London, 1979
J. A. Walker, Dynamical Systems and Evolution Equations, Plenum Press, New York, 1980
H. T. Banks and J. A. Burns, Hereditary control problems: numerical methods based on averaging approximations, SIAM J. Control Optim. 16, 169–208 (1978)
H. T. Banks, R. H. Fabiano, and Y. Wang, Estimation of Boltzmann damping coefficients in beam models, LCDS/CCS Report 88-13, Brown University, Providence, RI, 1988
H. T. Banks, R. H. Fabiano, Y. Wang, D. J. Inman, and H. Cudney, Spatial versus time hysteresis in damping mechanisms, 27th IEEE Conference on Decision and Control, 1988, pp. 1674–1677
J. A. Burns and R. H. Fabiano, Feedback control of a hyperbolic partial differential equation with viscoelastic damping, Control Theory Adv. Tech. 5, 157–188 (1989)
J. A. Burns, E. M. Cliff, Z. Y. Liu, and R. E. Miller, Control of a thermoviscoelastic system, 27th IEEE Conference on Decision and Control, 1988, pp. 1249–1252
J. A. Burns, Z. Y. Liu, and R. E. Miller, Approximations of thermoelastic and viscoelastic control systems, Numer. Funct. Anal. Optim. 12, 79–135 (1991)
C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal. 37, 297–308 (1979)
C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity, J. Differential Equations 7, 554–569 (1970)
C. M. Dafermos, Contraction semigroups and trend to equilibrium in continuum mechanics, Applications of Methods of Functional Analysis to Problems in Mechanics, Lecture Notes in Math., no. 503, Springer-Verlag, New York, 1976, pp. 295–306
W. Desch and R. K. Miller, Exponential stabilization of Volterra integrodifferential equations in Hilbert spaces, J. Differential Equations 70, 366–389 (1987)
R. H. Fabiano and K. Ito, Semigroup theory and numerical approximation for equations in linear viscoelasticity, SIAM J. Math. Anal. 21, 374–393 (1990)
R. H. Fabiano and K. Ito, Semigroup theory in linear viscoelasticity: Weakly and strongly singular kernels, Internat. Ser. Numer. Math., vol. 91, Birkhäuser, Basel, 1989, pp. 109–121
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, 1200–1234 (1988)
K. B. Hannsgen and R. L. Wheeler, Time delays and boundary feedback stabilization in one-dimensional viscoelasticity, Distributed Parameter Systems, Lecture Notes in Control and Inform. Sci., Springer, New York, 1987, pp. 136–152
T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1976
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983
H. Tanabe, Equations of Evolution, Pitman, London, 1979
J. A. Walker, Dynamical Systems and Evolution Equations, Plenum Press, New York, 1980
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Article copyright:
© Copyright 1994
American Mathematical Society
