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

We investigate the controlled harmonic oscillator \begin{equation*}\label{eq3.1}\tag{1} \ddot{\xi}+\xi=u, \end{equation*} where an external force (the control function) $u$ depends on the coordinate $\xi$, only. It can be shown that no ordinary (even nonlinear) feedback controls of the form $u=f(\xi(t))$ can asymptotically stabilize the solutions of the system \eqref{eq3.1}. However, one is able to make the system \eqref{eq3.1} asymptotically stable if one designs a special feedback control $u$ depending on $\xi(\cdot)$ which is called a hybrid feedback control. We demonstrate in the paper that the dynamics of a typical linear system of ordinary differential equations equipped with a linear hybrid feedback control possesses some irregular properties that dynamical systems without delay do not have. For example, solutions with different initial conditions may cross or even partly coincide. This proves that the hybrid dynamics cannot, in general, be described by a system of ordinary differential equations, neither linear, nor nonlinear, so that time-delays have to be incorporated into the system

Hybrid stabilization of planar linear systems with one-dimensional outputs

We consider a linear control system x'=Ax+Bu with output y=Cx, where x is two-dimensional, u,y are one-dimensional, and give necessary and sufficient conditions in order that it can be stabilized by a hybrid, linear feedback, where the action of the "switch" just depends on the sign of y. We also show, on these conditions, that the use of two control functions is enough for getting the goal