Free oscillations in systems with play (Q1287275)

The authors consider an automatic regulation system consisting of a linear unit and possessing a hysteresis nonlinearity (two-dimensional play) in the feedback loop. The dynamics of a linear unit is described by the differential equation \[ \dot{x} = Ax + By \] where \(A,B\) are \(2 \times 2\)-matrices. The play has the characteristic \(Z\) which is a compact convex subset of \(R^2\) with a smooth boundary. Its action obeys the discontinuous differential equation of the form \(\dot{y} = \nu (x-y)((\nu (x-y))^T \dot{x})^+\) for a.e. \(t \geq 0\) where \(a^+ = \max\{o,a\}\) and \(\nu (z) = 0\) if \(z \in int Z\) and \(\nu (z)\) is the unit vector of the external normal provided \(z \in \partial Z\). The conditions for the dissipativity of the system are given and the asymptotics of the phase trajectories are described. The invariant manifold is derived and a special class of play-boundary motions is chosen.