Scanning control of a vibrating string (Q1084063)

Let \(\Phi \in C^ 2({\mathbb{R}})\) be a function given a priori, and let y(x,t), \(\gamma_ i(t)\), \(i=1,...,N\) be a solution of the system of equations: \[ (F)\quad y_{tt}(x,t)=y_{xx}(x,t)+Ry(x,t)+\sum^{N}_{i=1}\Phi (x-\gamma_ i(t))y(x,t),\quad 0<x<1,\quad y=0\quad at\quad x=0,1; \] \[ \gamma_ i(t)+\lambda (\gamma_ i(t)-\gamma^*_ i)=-\int^{1}_{0}\Phi '(x- \gamma_ i(t))y^ 2(x,t)dx \] where \(\lambda >0\), \(\{\gamma^*_ i\}^ N_ 1\) are constants. Then, under some assumptions on \(\Phi\), it is shown that the phase points \((y,y_ t)(t)\) tends to equilibrium (0,0) as \(t\to \infty\), weakly in \(H^ 1_ 0(0,1)\times L_ 2(0,1)\). This feedback stabilization result is proved with the aid of nonlinear semigroup theory and qualitative methods obtained by \textit{J. M. Ball} [J. Differ. Equations 27, 224-265 (1978; Zbl 0376.35002)]. As a result, for a very large class of functions \(\Phi\) the \(\omega\)- limit set for a solution \((y,y_ t,\gamma)\) of (F) is a nonempty invariant subset of a certain manifold, while for a special class of translation invariant functions \(\Phi\) this \(\omega\)-limit sets shrinks to the origin; affirmative examples are given. In a final section a counterexample for \(\Phi\), supp \(\Phi\) compact, points out the difficulty in trying to approximate \(\Phi \in C_ 0^{\infty}({\mathbb{R}})\) by polynomials.