Destabilizing effects of small time delays on feedback-controlled descriptor systems
The following controlled descriptor system is considered \[ E\dot x(t)= Ax(t)+ Bu(t).\tag{1} \] Under certain conditions the following result is obtained: there exist sequences \((\varepsilon_j)\) and \((s_j)\) with \(\varepsilon_j> 0\), \(\varepsilon_j\to 0\), \(s_j\in \{s\in\mathbb{C}\mid\text{Re }s> 0\}\), \(\text{Im }\varepsilon_j\to \infty\), \(\text{Re }s_j\to \infty\), such that for any \(j\in \mathbb{N}\) the system (1) with a delayed feedback control \[ u(t)= Fx(t- \varepsilon_j)\tag{2} \] has the solution \[ x_j(t)= e^{s_jt} x_0,\quad x_0\neq 0, \] i.e., the system (1) with a feedback (2) is non-stable. Some examples are presented.
scientific article; zbMATH DE number 1116066
scientific article