ESAIM: Control, Optimisation and Calculus of Variations

Table of Contents - April 2006 - Volume 12, Issue 02  

Research Article

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

Khapalov, Alexander Y.

Department of Mathematics, Washington State University, Pullman, WA 99164-3113, USA; khapala@math.wsu.edu

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.

(Received March 31 2004)

(Revised December 9 2004)

(Online publication March 22 2006)

Key Words:

  • Semilinear wave equation;
  • approximate controllability;
  • multiplicative controls;
  • axial load;
  • damping.

Mathematics Subject Classification:

  • 93;
  • 35