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

Khapalov, Alexander Y.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 2, pp. 231-252.

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 $(y_0,y_1)\in H_0^1(0,1)\times L^2(0,1)$, by varying its axial load and the gain of damping, is dense in the “nonnegative” part of the subspace $L^2(0,1)\times \left\{0\right\}$ 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.

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