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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. (English) Zbl 1105.93011

ESAIM, Control Optim. Calc. Var. 12, 231-252 (2006).
Summary: 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 \( (0, \infty)\) with respect to the time-variable.

Cited in 8 Documents

MSC:

93B05 Controllability
35L70 Second-order nonlinear hyperbolic equations
74H45 Vibrations in dynamical problems in solid mechanics
74K05 Strings
74M05 Control, switches and devices (“smart materials”) in solid mechanics

Keywords:

semilinear wave equation; approximate controllability; multiplicative controls; axial load; damping
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References:

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