The global stability of two-dimensional systems for controlling angular orientation (Q2761357)

The author considers the system \(\dot\eta = -a\eta-f(\sigma)\), \(\dot\sigma = \beta\eta - bf(\sigma),\) where \(a\geq 0\), \(b>0\), \(\beta = 1-ab\neq 0\), \(\gamma\geq 0\), \(f(\sigma) = \varphi(\sigma) - \gamma\), \(\varphi(\sigma) = -\frac{M(\sigma)}{I}\), \(M(-0) = -M(+0),\) and at the discontinuity points \(\sigma = k\pi \) the function jumps are intervals \([M(2k\pi+0),M(2k\pi-0)] \) or \([M((2k+1)\pi-0),M((2k+1)\pi+0)]\), \([-\Delta,\Delta]\) is the inert zone. This two-dimensional system with an angular coordinate is able to describe the operations of a two-position automatic steering device, the plane rotations of a spacecraft controlled by angular orientation system, and a phase-lock control frequency system with a proportional-plus-integral filter. The following theorems are proved: a)~let \(\gamma=0\), then any solution of the above system tends to an equilibrium state; b) let \(\gamma>0\), \(a=0\), then there exist a positive number \(\varepsilon\) and solutions \(\eta(t)\), \(\sigma(t)\) of the above system such that \(\eta(t)\geq \varepsilon \) for all \(t\geq 0\).