Conditions of stabilization of nonlinear systems by a harmonic external action (Q1914131)

The author considers the \(n\)-dimensional system \[ \dot x= Px+ q(\varphi(\sigma)+ \alpha \sin \omega_0 t),\quad \sigma= r^t x, \] where \(P\) is a constant matrix, \(q\) and \(r\) are \(n\)-dimensional vectors, \(\varphi(\sigma)\) is the nonlinear feedback characteristic and \(\alpha\), \(\omega_0\) denote respectively amplitude and frequency of the external force. It is assumed that \(P\) is Hurwitz and that \(\varphi\), \(\varphi'\) satisfy certain boundedness conditions. However, in general, the graph of \(\varphi\) does not satisfy the well-known sector condition for absolute stability, so that periodic steady state solutions (with the same frequency as the forcing term) may arise. The author shows that under certain conditions involving the parameter of the system and the forcing term, for each pair of solutions \(x_1(t)\), \(x_2(t)\) one has \(\lim_{t\to + \infty} (x_1(t)- x_2(t))= 0\).