Lyapunov stability of a class of hybrid dynamic systems (Q1962011)

The authors consider the ``hybrid system'' \[ \dot x= g(x,m(\tau)) \] with \(m(\tau)\) the ``discrete state'' of the system and \(x\) the continuous state. When the trajectory meets the set \(\{x(\tau): (x(\tau), m(\tau))= 0\}\) some switching will occur \[ x(\tau_k^+)= h(x(\tau_k^-), m(\tau_k^-)), \qquad m(\tau_k^+)= \psi(x(\tau_k^-), m(\tau_k^-)). \] Stability of the invariant sets of such systems is considered using a kind of Lyapunov function interpreted as an abstract measure of systems potential energy.