Reachability of Nonnegative Equilibrium States for the Semilinear Vibrating String

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 nonzero initial state (y₀,y₁) εH¹ ₀ (0,1)×L²(0,1), by varying its axial load and the gain of damping, is dense in the “gnonnegative” part of the subspace L²(0,1)×{0} of L²(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.