Abstract
It is shown that a general isotropic viscoelastic solid with non vanishing Newtonian viscosity is never exactly controllable using L2-boundary controls. For some models it is known that even spectral controllability does not hold. Here we show, thereby extending results obtained in Leugering and Schmidt [10], that the general model is approximatively controllable under some reasonable assumptions.
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References
Baumeister J (1983) Boundary control of an integrodifferential equation. J. Math Anal Appl 93: 550–570
Da Prato G, Ianelli M (1980) Linear integro-differential equations in Banach spaces. Red Sem Mat Univ Padova 62: 207–219
Da Prato G, Lunardi A (1987) Solvability of the real line of a class of linear Volterra integrodifferential equations of parabolic type. Universita Degli Studi di Pisa Preprint Nr 178
Duvaut G, Lions JL (1972) Les Inéquations en Mécanique et en Physique. Dunod Paris
Gaiduk SI (1977) Some problems related to the theory of the action of transverse impulse on a rod. Differential equations 13/II: 854–861
Hörmander L (1963) Linear Partial Differential Operators. Springer Verlag
Leugering G (1986) Boundary controllability of a viscoelastic beam. Applicable Analysis 23: 119–137
Leugering G (1987) Exact boundary controllability of an integrodifferential equation. Appl Math Optim 15: 223–250
Leugering G (1987) On boundary controllability of Volterra integrodifferential equations in Hilbert spaces. To appear in the proceedings of the Conference on Control of Distributed Parameter Systems, Vorau, Austria 1986
Leugering G, Schmidt EJPG (1987) Boundary control of a vibrating plate with internal damping. Submitted
Lions JL (1986) Exact controllability, stabilization and perturbations for distributed systems. The John von Neumann Lecture, SIAM National Meeting, Boston, USA 1986
Lunardi A (1985) Laplace transforms methods in integrodifferential equations. J Integral Eq 10 (1–3) suppl 185–211
MacCamy RC, Mizel VJ, Seidman TI (1968) Approximate boundary controllability for the heat equation. J Math Anal Appl 23: 699–703
Miller RK, Wheeler RL (1977) Asymptotic behavior for a linear Volterra integral equation in Hilbert space. J Diff Eq 23: 270–284
Nečas J (1967) Les méthodes directes en théorie des équation elliptiques. Masson et Cie, Paris
Nerain A, Joseph DD (1982) Linearized dynamics for step jumps of velocity and displacement of shearing flows of a simple fluid. Rheol Acta 21: 228–250
Prüss J (1987) Positivity and regularity of hyperbolic Volterra equations in Banach spaces. Math Annalen to appear
Renardy M (1982) Some remarks on propagation and nonpropagation of discontinuities in linearly viscoelastic liquids. Rheol Acta 21: 251–254
Russell DL (1985) Mathematical models for the elastic beam and their control theoretic implications. Semigroup Theory and Applications (Brezis et at Eds) Longman New York
Schmidt EJPG, Weck N (1978) On the boundary behavior of solutions to elliptic and parabolic equations — with applications to boundary control for parabolic equations. SIAM Control Optim 4: 593–598
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© 1989 International Federation for Information Processing
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Leugering, G. (1989). On boundary controllability of viscoelastic systems. In: Bermúdez, A. (eds) Control of Partial Differential Equations. Lecture Notes in Control and Information Sciences, vol 114. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0002592
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DOI: https://doi.org/10.1007/BFb0002592
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