Hybrid dynamical systems vs. ordinary differential equations: Examples of a "pathological" behavior

with a control u = u(y). Here u again depends only on the output y = ξ. It can be shown (see e.g. [1]) that there is no output feedback control of the form u = f(ξ) = f(ξ(t)) that makes the system (2) asymptotically stable. Therefore, it was suggested in [1] to use hybrid feedback controls (abbr. HFC), which indeed can stabilize the system (2). The idea used in [1] can be roughly described as follows. We incorporate a discrete device (an automaton) into the considered system ( a plant). The device is able to switch on and off certain control functions at certain instances. The time interval between two consecutive switchings depends on the last observation of ξ. As it was demonstrated in [1], careful choice of design procedure and switching instances provides asymptotic stability of the system (2). The discrete nature of hybrid outputs makes their practical implementation simpler. More results on stabilization of linear and nonlinear systems via HFC with a finite number of automata’s locations are available (see e.g. [2], [3], [4], [5], [6], [8], [10]). In [7] it was proved that it is possible to stabilize an arbitrary linear system by using HFC with infinitely many locations.