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Reachability of nonnegative equilibrium states for the semilinear vibrating string by varying its axial load and the gain of damping

Published online by Cambridge University Press:  22 March 2006

Alexander Y. Khapalov
Alexander Y. Khapalov*
Affiliation:
Department of Mathematics, Washington State University, Pullman, WA 99164-3113, USA; khapala@math.wsu.edu
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Abstract

We show that the set of nonnegative equilibrium-like states, namely, like $ (y_d, 0) $ of the semilinear vibrating string that can be reached from any non-zero initial state $ (y_0, y_1) \in H^1_0 (0,1) \times L^2 (0,1)$, by varying its axial load and the gain of damping, is dense in the “nonnegative” part of the subspace $ L^2 (0,1) \times \{0\} $ of $ L^2 (0,1) \times H^{-1} (0,1)$. Our main results deal with nonlinear terms which admit at most the linear growth at infinity in $ \; y \; $ and satisfy certain restriction on their total impact on (0,∞) with respect to the time-variable.

Keywords

Semilinear wave equationapproximate controllabilitymultiplicative controlsaxial loaddamping.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 12 , Issue 2 , April 2006 , pp. 231 - 252
Copyright
© EDP Sciences, SMAI, 2006

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