Almost Global Finite-time Stability of Invariant Sets

A relation between finite-time stability and Rantzer's density function has been presented for both discrete- and continuous-time system. We show that the existence of an integrable Rantzer function (also called Lyapunov density or Lyapunov measure) implies the convergence of almost all trajectories to an invariant set in finite time. The proofs utilise the duality between Frobenious-Perron operator and Koopman operator, as well as Rantzer's lemma for the evolution of densities. To illustrate the theoretical results, some illustrative examples are presented. Additionally, a transformation that removes the integrability assumption has been addressed which makes the problem of the construction of a Rantzer function numerically tractable.