Stability of a linear integro-differential equation with periodic coefficients

Authors:
Aleksey D. Drozdov and Michael I. Gil

Journal:
Quart. Appl. Math. **54** (1996), 609-624

MSC:
Primary 34K20; Secondary 45J05, 45M10, 73F15

DOI:
https://doi.org/10.1090/qam/1417227

MathSciNet review:
MR1417227

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Abstract: Stability of a linear integro-differential equation with periodic coefficients is studied. Such an equation arises in the dynamics of thin-walled viscoelastic elements of structures under periodic compressive loading. The equation under consideration has a specific peculiarity which makes its analysis difficult: in the absence of the integral term it is only stable, but not asymptotically stable. Therefore, in order to derive stability conditions we have to introduce some specific restrictions on the behavior of the kernel of the integral operator. These restrictions are taken from the analysis of the relaxation measures for a linear viscoelastic material.

**[1]**N. Kh. Arutyunyan, A. D. Drozdov, and V. E. Naumov,*Mechanics of Growing Viscoelastoplastic Bodies*, Nauka, Moscow, 1987 (in Russian)**[2]**L. Kh. Belen'kaya and V. I. Judovich,*Stability of a viscoelastic rod under a periodic load*, Mech. Solids**13**, no. 6, 128-133 (1978)**[3]**F. Bloom,*On the existence of solutions to nonstrictly hyperbolic problems in nonlinear viscoelasticity*, Applicable Analysis**17**, 115-133 (1984)**[4]**V. V. Bolotin,*Dynamic Stability of Elastic Systems*, Holden Day, San Francisco, 1964**[5]**T. A. Burton,*Volterra Integral and Differential Equations*, Academic Press, New York, 1983**[6]**G. Cederbaum and M. Mond,*Stability properties of a viscoelastic column under a periodic force*, Trans. Amer. Soc. Mech. Eng. J. Appl. Mech.**59**, 16-19 (1992)**[7]**B. D. Coleman and D. Hill,*On the stability of certain motions of incompressible materials with memory*, Arch. Rational Mech. Anal.**30**, no. 3, 197-224 (1968)**[8]**C. Corduneanu and V. Lakshmikantham,*Equations with infinite delay. A survey*, Nonlinear Anal.**4**, 831-877 (1980)**[9]**R. M. Christensen,*Theory of Viscoelasticity. An Introduction*, Academic Press, New York, 1982**[10]**C. M. Dafermos,*An abstract Volterra equation with applications to linear viscoelasticity*, J. Differential Equations**7**, 554-569 (1970)**[11]**C. M. Dafermos and J. A. Nohel,*Energy methods for nonlinear hyperbolic Volterra integrodifferential equations*, Comm. Partial Differential Equations**4**, 219-278 (1979)**[12]**A. Drozdov,*Stability of viscoelastic shells under periodic and stochastic loading*, Mech. Res. Comm.**20**, no. 6, 481-486 (1993)**[13]**A. D. Drozdov and V. B. Kolmanovskii,*Stochastic stability of viscoelastic bars*, Stochastic Anal. Appl.**10**, no. 3, 265-276 (1992)**[14]**A. D. Drozdov, V. B. Kolmanovskii, and P. A. Velmisov,*Stability of Viscoelastic Systems*, Saratov Univ. Press, Saratov, 1991 (in Russian)**[15]**N. P. Erugin,*Linear Systems of Ordinary Differential Equations with Periodic and Quasi-Periodic Coefficients*, Academic Press, New York, 1966**[16]**Kh. Eshmatov and P. Kurbanov,*Parametric oscillations in a viscoelastic rod with a nonlinear inheritance characteristic*, Appl. Math. Mech.**39**, no. 4 (1975)**[17]**R. M. Evan-Iwanowski,*Resonance Oscillations in Mechanical Systems*, Elsevier, New York, 1976**[18]**M. Fabrizio and B. Lazzari,*On the existence and the asymptotic stability of solutions for linearly viscoelastic solids*, Arch. Rational Mech. Anal.**116**, no. 2, 139-152 (1991)**[19]**M. Fabrizio and A. Morro,*Mathematical Problems in Linear Viscoelasticity*, SIAM Studies in Applied Mathematics**12**(1992)**[20]**G. Gripenberg, S. O. Londen, and O. Staffans,*Volterra Integral and Functional Equations*, Cambridge Univ. Press, Cambridge, 1990**[21]**J. K. Hale,*Theory of Functional Differential Equations*, Springer-Verlag, New York, 1977**[22]**G. Herrmann,*Dynamic Stability of Structures*, Pergamon Press, Oxford, 1967**[23]**R. A. Horn and C. R. Johnson,*Matrix Analysis*, Cambridge Univ. Press, Cambridge, 1985**[24]**V. B. Kolmanovskii and V .R. Nosov,*Stability of Functional Differential Equations*, Academic Press, New York, 1986**[25]**R. C. MacCamy,*A model for one-dimensional nonlinear viscoelasticity*, Quart. Appl. Math.**35**, no. 1, 1-19 (1977)**[26]**I. G. Malkin,*Some Problems of the Theory of Nonlinear Oscillations*, Gostechizdat, Moscow, 1956 (in Russian)**[27]**V. I. Matyash,*Dynamic stability of a hinged viscoelastic bar*, Mech. Polymers**2**, 293-300 (1967)**[28]**N. W. Mclachlan,*Theory and Applications of Mathieu Functions*, Dover Publications, New York, 1964**[29]**M. Renardy, W. J. Hrusa, and J. A. Nohel,*Mathematical Problems in Viscoelasticity*, Longmans Press, Essex, 1987**[30]**K. K. Stevens,*On the parametric excitation of a viscoelastic column*, AIAA J.**12**, 2111-2116 (1966)**[31]**A. Stokes,*A Floquet theory for functional differential equations*, Proc. Nat. Acad. Sci. USA**48**, 1330-1334 (1962)**[32]**V. A. Yakubovich and V. M. Starzhinskii,*Linear Differential Equations with Periodic Coefficients*, Vols. 1, 2, John Wiley, New York, 1975

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DOI:
https://doi.org/10.1090/qam/1417227

Article copyright:
© Copyright 1996
American Mathematical Society