Finite time stability of nonlinear systems

Finite time stability is defined for non autonomous non linear systems. Starting with the proof of an old result from Haimo which gives necessary and sufficient condition for autonomous scalar systems, then we extend this result to n-dimensional systems through the use of Liapunov functions and (here the obtained results are less restrictives). For this we introduce the notion of upper bounded Lyapunov pair (respectively lower bounded Lyapunov pair) to obtain sufficient (respectively necessary) conditions for finite time stability.