Stabilization of bilinear oscillating systems with two degrees of freedom (Q1280337)

The authors treat the bilinear oscillating system with two degrees of freedom \[ \begin{aligned} &\ddot x_i + (\omega^2_i + u_1)x_i = u_2,\qquad i = 1,2,\\ &u_i(\cdot)\in U_i = \{u_i(\cdot)\in KC^1\: |u_i(t)|\leq \nu_i\}. \end{aligned}\tag{1} \] The piecewise linear functions of the scalar argument \(s\) \[ u_i = \nu_i \widetilde{\operatorname{sign}} s_i = \begin{cases} \frac{\nu_i}{\delta} s_i &\quad\text{for }|s_i|\leq\delta,\\ \nu_i \operatorname{sign} s_i &\quad\text{for } |s_i|>\delta \end{cases} \] are used as the controls. The problem of system (1) stabilization is studied for two cases: \(\omega_1\neq\omega_2\) and \(\omega_1=\omega_2=\omega_0\).