Observability and stabilization of the vibrating string equipped with bouncing point sensors and actuators (Q2769912)

This paper deals with solving the problems. The first one is the Dirichlet problem NEWLINE\[NEWLINEw_{tt}=w_{xx},\;(x,t)\in(0,1) \times(0,\infty)NEWLINE\]NEWLINE NEWLINE\[NEWLINEw(0,t)= w(1,t)=0,\;w(x,0)= w_0\in H^1_0(0,1),\;w_t(x,0)= w_1\in L^2(0,1)NEWLINE\]NEWLINE with \((w_0, w_1)\) unknown, but assuming that there is known \(w(s(t),t)\) for some interval \((0,T)\), \(s(t)\) being given on \((0,\infty)\). The second problem is stability of the equilibrium solution for the problem NEWLINE\[NEWLINEy_{tt}= y_{xx}-k y_t\delta \bigl(x-s(t) \bigr),\;(x,t)\in(0,1) \times\bigl( (0,\infty) \setminus \bigcup^\infty_1 (t_i-\delta_i, t_i +\delta_i)\bigr)NEWLINE\]NEWLINE NEWLINE\[NEWLINEy_{tt}= y_{xx}-\widehat ky_t\chi_0 (x, t),\;(x,t)\in (0,1)\times \bigcup^\infty_1 (t_i-\delta_i, t_i+\delta_i)NEWLINE\]NEWLINE NEWLINE\[NEWLINEy(0, t)= y(1,t)=0,\;y(x,0)= y_0\in H^1_0(0,1),\;y_t(x,0)= y_1\in L^2(0,1)NEWLINE\]NEWLINE with \(\chi_0\) the characteristic function of the given set \(D=((0,1) \times(0, \infty)\cap (\bigcup^\infty_1 ((x_i-\nu_i, x_i+\nu_i) \times(t_i- \delta_i, t_i+ \delta_i))))\), \(x_i=s(t_i)\), \(\nu_i>0\), \(0<\delta_i <\min(t_i-t_{i-1}, t_{i+1}-t_i)\).