Reachability of Nonnegative Equilibrium States for the Semilinear Vibrating String by Varying Its Axial Load and the Gain of Damping

Abstract. We show that the set of nonnegative equilibrium-like states, namely, like (yd, 0) of the semilinear vibrating string that can be reached from any non-zero initial state (y0, y1) ∈ H 0 (0, 1) × L(0, 1), by varying its axial load and the gain of damping, is dense in the “nonnegative” part of the subspace L(0, 1) × {0} of L(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.