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

Alexander Y. Khapalov

ESAIM: Control, Optimisation and Calculus of Variations (2006)

  • Volume: 12, Issue: 2, page 231-252
  • ISSN: 1292-8119

Abstract

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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 ) ∈ H 0 1 ( 0 , 1 ) × 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 ) × { 0 } of L 2 ( 0 , 1 ) × 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.

How to cite

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Khapalov, Alexander Y.. "Reachability of nonnegative equilibrium states for the semilinear vibrating string by varying its axial load and the gain of damping." ESAIM: Control, Optimisation and Calculus of Variations 12.2 (2006): 231-252. <http://eudml.org/doc/249674>.

@article{Khapalov2006,
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. },
author = {Khapalov, Alexander Y.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Semilinear wave equation; approximate controllability; multiplicative controls; axial load; damping. ; semilinear wave equation; damping},
language = {eng},
month = {3},
number = {2},
pages = {231-252},
publisher = {EDP Sciences},
title = {Reachability of nonnegative equilibrium states for the semilinear vibrating string by varying its axial load and the gain of damping},
url = {http://eudml.org/doc/249674},
volume = {12},
year = {2006},
}

TY - JOUR
AU - Khapalov, Alexander Y.
TI - Reachability of nonnegative equilibrium states for the semilinear vibrating string by varying its axial load and the gain of damping
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2006/3//
PB - EDP Sciences
VL - 12
IS - 2
SP - 231
EP - 252
AB - 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.
LA - eng
KW - Semilinear wave equation; approximate controllability; multiplicative controls; axial load; damping. ; semilinear wave equation; damping
UR - http://eudml.org/doc/249674
ER -

References

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Citations in EuDML Documents

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  1. Carlos Castro, Exact controllability of the 1-d wave equation from a moving interior point
  2. Luis Alberto Fernández, Alexander Yuri Khapalov, Controllability properties for the one-dimensional Heat equation under multiplicative or nonnegative additive controls with local mobile support
  3. Alexander Khapalov, Source localization and sensor placement in environmental monitoring
  4. Piermarco Cannarsa, Patrick Martinez, Judith Vancostenoble, Null controllability of the heat equation in unbounded domains by a finite measure control region
  5. Piermarco Cannarsa, Patrick Martinez, Judith Vancostenoble, Null controllability of the heat equation in unbounded domains by a finite measure control region

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