Wellposedness and stabilizability of a viscoelastic equation in energy space
Author:
Olof J. Staffans
Journal:
Trans. Amer. Math. Soc. 345 (1994), 527575
MSC:
Primary 45K05; Secondary 34K30, 35Q72, 73F05, 73F15, 93D15
MathSciNet review:
1264153
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Abstract: We consider the wellposedness and exponential stabilizability of the abstract Volterra integrodifferential system in ilbeubert space. In a typical viscoelastic interpretation of this equation one lets v represent velocit, acceleratio , stres, the divergence of the stres, pure viscosity (usually equal to zero) Dv the time derivative of the strain, and a the linear stress relaxation modulus of the material. The problems that we treat are onedimensional in the sense that we require a to be scalar. First we prove wellposedness in a new semigroup setting, where the history component of the state space describes the absorbed energy of the system rather than the history of the function v. To get the wellposedness we need extremely weak assumptions on the kernel; it suffices if the system is "passive", i.e., a is of positive type; it may even be a distribution. The system is exponentially stabilizable with a finite dimensional continuous feedback if and only if the essential growth rate of the original system is negative. Under additional assumptions on the kernel we prove that this is indeed the case. The final part of the treatment is based on a new class of kernels. These kernels are of positive type, but they need not be completely monotone. Still, they have many properties similar to those of completely monotone kernels, and a number of results that have been proved earlier for completely monotone kernels can be extended to the new class.
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 [2]
 , Spectral resolution for integrodifferential equations, Proc. 28th IEEE Conf. Decision and Control (Tampa, FL), 1989, pp. 151154. MR 1038925 (91a:47054)
 [3]
 Wolfgang Desch and Richard K. Miller, Exponential stabilization of Volterra integrodifferential equations in Hilbert space, J. Differential Equations 70 (1987), 366389. MR 915494 (89a:45025)
 [4]
 , Exponential stabilization of Volterra integral equations with singular kernels, J. Integral Equations Appl. 1 (1988), 397433. MR 1003703 (90f:93050)
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 L. Gerhart, Spectral theory for contraction semigroups on Hilbert spaces, Trans. Amer. Math. Soc. 236 (1978), 385394. MR 0461206 (57:1191)
 [6]
 Gustaf Gripenberg, StigOlof Londen, and Olof J. Staffans, Volterra integral and functional equations, Cambridge Univ. Press, Cambridge and New York, 1990.
 [7]
 Kenneth B. Hannsgen and Robert L. Wheeler, Behavior of the solution of a Volterra equation as a parameter tends to infinity, J. Integral Equations 7 (1984), 229237. MR 770149 (86b:45004)
 [8]
 Daniel D. Joseph, Fluid dynamics of viscoelastic liquids, SpringerVerlag, New York and Berlin, 1990. MR 1051193 (91d:76003)
 [9]
 Paul Koosis, Introduction to spaces, Cambridge Univ. Press, Cambridge, 1980. MR 565451 (81c:30062)
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 Günter Leugering, A decomposition method for integropartial differential equations and applications, J. Math. Pures Appl. 71 (1992), 561587. MR 1193609 (93i:45012)
 [11]
 , Spectral decomposition of Volterra integrodifferential equations with application to the control of a viscoelastic solid, Proc. 28th IEEE Conf. Decision and Control (Tampa, FL), 1989, pp. 22822285.
 [12]
 , On control and stabilization of a rotating beam by applying moments to the base only, Optimal Control of Partial Differential Equations, SpringerVerlag, 1991, pp. 182191.
 [13]
 A. Pazy, Semigroups of linear operators and applications to partial differential equations, SpringerVerlag, Berlin, 1983.
 [14]
 Allen C. Pipkin, Lectures on viscoelasticity theory, 2nd ed., SpringerVerlag, Berlin, 1986.
 [15]
 Jan Prüss, On the spectrum of semigroups, Trans. Amer. Math. Soc. 284 (1984), 847857.
 [16]
 , Evolutionary integral equations and applications, Monographs Math., vol. 87, Birkhäuser, Basel and Boston, 1993.
 [17]
 Michael Renardy, William J. Hrusa, and John A. Nohel, Mathematical problems in viscoelasticity, Longman Scientific, Burnt Mill, 1987. MR 919738 (89b:35134)
 [18]
 Walter Rudin, Real and complex analysis, McGrawHill, New York, 1974. MR 0344043 (49:8783)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002994719941264153X
PII:
S 00029947(1994)1264153X
Keywords:
Passive system,
essential spectrum,
essential growth rate,
spectrum detrmined growth,
absorbed energy,
internal energy,
relaxation modulus
Article copyright:
© Copyright 1994
American Mathematical Society
