An inverse problem for the linear viscoelastic Kirchhoff plate
Authors:
Cecilia Cavaterra and Maurizio Grasselli
Journal:
Quart. Appl. Math. 53 (1995), 9-33
MSC:
Primary 73K10; Secondary 35R30, 45K05, 47N20, 73F15
DOI:
https://doi.org/10.1090/qam/1315445
MathSciNet review:
MR1315445
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Abstract: We consider the linear Kirchhoff plate model subject to a viscoelastic damping of integral type. The damping term contains a time-dependent convolution kernel accounting for the long-range memory effects. The mechanical behavior of the plate is influenced by this memory function which is a priori unknown. That leads us to consider the question of identifying it. We study this kind of inverse problem by using the integrodifferential evolution equation governing the vertical deflection of the plate and coupling it with a set of overposed initial and boundary conditions. The problem obtained is then reduced to a nonlinear Volterra integral equation of second kind for the unknown memory kernel. Then, via the Contraction Principle, we prove local (in time) existence and uniqueness results. In addition, we show the Lipschitz continuous dependence upon the data. These results also apply to a viscoelastic beam model.
H. T. Banks, R. H. Fabiano, and Y. Wang, Inverse problem techniques for beams with tip body and time hysteresis damping, Mat. Aplic. Comput. (2) 8, 101–118 (1989)
A. Ben-Menahem and S. J. Singh, Seismic Waves and Sources, Springer-Verlag, New York, 1981
J. M. Carcione, Wave propagation in anisotropic linear viscoelastic media: theory and simulated wavefields, Geophys. J. 101, 739–750 (1990)
G. Caviglia and A. Morro, On the modelling of dissipative solids, Meccanica 25, 124–127 (1990)
G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin-Heidelberg, 1976
G. Fichera, I difficili rapporti fra l’Analisi funzionale e la Fisica Matematica, Rend. Mat. Acc. Lincei (9) 1, 161–170 (1990)
J. M. Golden and G. A. C. Graham, Boundary Value Problems in Linear Viscoelasticity, SpringerVerlag, Berlin-Heidelberg, 1988
W. N. Findley, J. S. Lai, and K. Onaran, Creep and relaxation of nonlinear viscoelastic materials, Series in Appl. Math. and Mech., vol. 18, North-Holland, Amsterdam, 1976
M. Grasselli, S. I. Kabanikhin, and A. Lorenzi, An inverse hyperbolic integrodifferential problem arising in Geophysics, I, Sibirsk. Mat. Zh. 33, 58–68 (1992) (Russian)
M. Grasselli, S. I. Kabanikhin, and A. Lorenzi, An inverse hyperbolic integrodifferential problem arising in Geophysics, II, Nonlinear Anal. TMA 15, 283–298 (1990)
M. Grasselli, An identification problem for an abstract linear hyperbolic integrodifferential equation with applications, J. Math. Anal. Appl. 171, 27–60 (1992)
M. Grasselli, An inverse problem in three-dimensional linear thermoviscoelasticity of Boltzmann type, Ill-Posed Problems in Natural Sciences (A. N. Tikhonov, ed.), VSP, Zeist, 1992, p. 284
M. Gurtin and E. Sternberg, On the linear theory of viscoelasticity, Arch. Rational Mech. Anal. 11, 291–356 (1962)
W. J. Hrusa, J. A. Nohel, and M. Renardy, Mathematical problems in viscoelasticity, Pitman Monographs and Surveys in Pure and Applied Math., vol. 35, Longman, Harlow, 1987
J. E. Lagnese and J.-L. Lions, Modelling, analysis, and control of thin plates, Coll. RMA 6, Masson, Paris, 1988
J. E. Lagnese, Asymptotic energy estimates for Kirchhoff plates subject to weak viscoelastic damping, Control and Estimation of Distributed Parameter Systems (F. Kappel, K. Kunisch, and W. Schappacher, eds.), Internat. Series Num. Math., vol 91, Birkhäuser-Verlag, Berlin, 1989, p. 211
J. E. Lagnese, Boundary stabilization of thin plates, SIAM Stud. Appl. Math., vol. 10, SIAM, Philadelphia, 1989
J. E. Lagnese, The reachability problem for thermoelastic plates, Arch. Rational Mech. Anal. 112, 223–267 (1990)
I. Lasiecka, Controllability of a viscoelastic Kirchhoff plate, Control and Estimation of Distributed Parameter Systems (F. Kappel, K. Kunisch, and W. Schappacher, eds.), Internat. Series Num. Math., vol. 91, Birkhäuser-Verlag, Berlin, 1989, p. 237
G. Leugering, Exact boundary controllability of an integro-differential equation, Appl. Math. Optim. 15, 223–250 (1987)
G. Leugering, On boundary feedback stabilisability of a viscoelastic beam, Proc. Roy. Soc. Edinburgh Sect. A 114, 57–69 (1990)
J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, Berlin, 1972
A. Lorenzi, An identification problem related to a nonlinear hyperbolic integrodifferential equation, Nonlinear Anal. TMA 22, 21–44 (1994)
A. E. Osokin and Iu. V. Suvorova, Nonlinear governing equation of a hereditary medium and methodology of determining its parameters, J. Appl. Math. Mech. 42, 1214–1222 (1978) (English transl. of Prikl. Mat. Mekh.)
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Appl. Math. Sci., vol. 44, Springer-Verlag, New York, 1983
A. C. Pipkin, Lectures on viscoelasticity, Appl. Math. Sci., vol. 7, Springer-Verlag, New York, 1972
Yu. N. Rabotnov, Elements of hereditary solid mechanics, Mir Publishers, Moscow, 1980
J. L. Sackman, Prediction and identification in viscoelastic wave propagation, Wave Propagation in Viscoelastic Media (F. Mainardi, ed.), Pitman Res. Notes in Math., vol. 52, Longman, Harlow, 1987
T. A. Tobias and Yu. K. Englebrecht, Inverse problems for evolution equations of the integrodifferential type, J. Appl. Math. Mech. 49, 519–524 (1985) (English transl. of Prikl. Mat. Mekh.)
Y. Yanno, On an inverse problem for a hyperbolic equation, Uch. Zap. Tartu Gos. Univ. 672, 40–46 (1984) (Russian)
H. T. Banks, R. H. Fabiano, and Y. Wang, Inverse problem techniques for beams with tip body and time hysteresis damping, Mat. Aplic. Comput. (2) 8, 101–118 (1989)
A. Ben-Menahem and S. J. Singh, Seismic Waves and Sources, Springer-Verlag, New York, 1981
J. M. Carcione, Wave propagation in anisotropic linear viscoelastic media: theory and simulated wavefields, Geophys. J. 101, 739–750 (1990)
G. Caviglia and A. Morro, On the modelling of dissipative solids, Meccanica 25, 124–127 (1990)
G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin-Heidelberg, 1976
G. Fichera, I difficili rapporti fra l’Analisi funzionale e la Fisica Matematica, Rend. Mat. Acc. Lincei (9) 1, 161–170 (1990)
J. M. Golden and G. A. C. Graham, Boundary Value Problems in Linear Viscoelasticity, SpringerVerlag, Berlin-Heidelberg, 1988
W. N. Findley, J. S. Lai, and K. Onaran, Creep and relaxation of nonlinear viscoelastic materials, Series in Appl. Math. and Mech., vol. 18, North-Holland, Amsterdam, 1976
M. Grasselli, S. I. Kabanikhin, and A. Lorenzi, An inverse hyperbolic integrodifferential problem arising in Geophysics, I, Sibirsk. Mat. Zh. 33, 58–68 (1992) (Russian)
M. Grasselli, S. I. Kabanikhin, and A. Lorenzi, An inverse hyperbolic integrodifferential problem arising in Geophysics, II, Nonlinear Anal. TMA 15, 283–298 (1990)
M. Grasselli, An identification problem for an abstract linear hyperbolic integrodifferential equation with applications, J. Math. Anal. Appl. 171, 27–60 (1992)
M. Grasselli, An inverse problem in three-dimensional linear thermoviscoelasticity of Boltzmann type, Ill-Posed Problems in Natural Sciences (A. N. Tikhonov, ed.), VSP, Zeist, 1992, p. 284
M. Gurtin and E. Sternberg, On the linear theory of viscoelasticity, Arch. Rational Mech. Anal. 11, 291–356 (1962)
W. J. Hrusa, J. A. Nohel, and M. Renardy, Mathematical problems in viscoelasticity, Pitman Monographs and Surveys in Pure and Applied Math., vol. 35, Longman, Harlow, 1987
J. E. Lagnese and J.-L. Lions, Modelling, analysis, and control of thin plates, Coll. RMA 6, Masson, Paris, 1988
J. E. Lagnese, Asymptotic energy estimates for Kirchhoff plates subject to weak viscoelastic damping, Control and Estimation of Distributed Parameter Systems (F. Kappel, K. Kunisch, and W. Schappacher, eds.), Internat. Series Num. Math., vol 91, Birkhäuser-Verlag, Berlin, 1989, p. 211
J. E. Lagnese, Boundary stabilization of thin plates, SIAM Stud. Appl. Math., vol. 10, SIAM, Philadelphia, 1989
J. E. Lagnese, The reachability problem for thermoelastic plates, Arch. Rational Mech. Anal. 112, 223–267 (1990)
I. Lasiecka, Controllability of a viscoelastic Kirchhoff plate, Control and Estimation of Distributed Parameter Systems (F. Kappel, K. Kunisch, and W. Schappacher, eds.), Internat. Series Num. Math., vol. 91, Birkhäuser-Verlag, Berlin, 1989, p. 237
G. Leugering, Exact boundary controllability of an integro-differential equation, Appl. Math. Optim. 15, 223–250 (1987)
G. Leugering, On boundary feedback stabilisability of a viscoelastic beam, Proc. Roy. Soc. Edinburgh Sect. A 114, 57–69 (1990)
J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, Berlin, 1972
A. Lorenzi, An identification problem related to a nonlinear hyperbolic integrodifferential equation, Nonlinear Anal. TMA 22, 21–44 (1994)
A. E. Osokin and Iu. V. Suvorova, Nonlinear governing equation of a hereditary medium and methodology of determining its parameters, J. Appl. Math. Mech. 42, 1214–1222 (1978) (English transl. of Prikl. Mat. Mekh.)
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Appl. Math. Sci., vol. 44, Springer-Verlag, New York, 1983
A. C. Pipkin, Lectures on viscoelasticity, Appl. Math. Sci., vol. 7, Springer-Verlag, New York, 1972
Yu. N. Rabotnov, Elements of hereditary solid mechanics, Mir Publishers, Moscow, 1980
J. L. Sackman, Prediction and identification in viscoelastic wave propagation, Wave Propagation in Viscoelastic Media (F. Mainardi, ed.), Pitman Res. Notes in Math., vol. 52, Longman, Harlow, 1987
T. A. Tobias and Yu. K. Englebrecht, Inverse problems for evolution equations of the integrodifferential type, J. Appl. Math. Mech. 49, 519–524 (1985) (English transl. of Prikl. Mat. Mekh.)
Y. Yanno, On an inverse problem for a hyperbolic equation, Uch. Zap. Tartu Gos. Univ. 672, 40–46 (1984) (Russian)
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© Copyright 1995
American Mathematical Society
