  In this work, we study finite-time stability of switched and hybrid systems
in the presence of unstable modes. We present sufficient conditions in terms of
multiple Lyapunov functions for the origin of the system to be finite time
stable. More specifically, we show that even if the value of the Lyapunov
function increases in between two switches, i.e., if there are unstable modes
in the system, finite-time stability can still be guaranteed if the finite time
convergent mode is active long enough. In contrast to earlier work where the
Lyapunov functions are required to be decreasing during the continuous flows
and non-increasing at the discrete jumps, we allow the Lyapunov functions to
increase \emph{both} during the continuous flows and the discrete jumps. As
thus, the derived stability results are less conservative compared to the
earlier results in the related literature, and in effect allow the hybrid
system to have unstable modes. Then, we illustrate how the proposed finite-time
stability conditions specialize for a class of switched systems, and present a
method on the synthesis of a finite-time stabilizing switching signal for
switched linear systems. As a case study, we design a finite-time stable output
feedback controller for a linear switched system, in which only one of the
modes is controllable and observable. Numerical example demonstrates the
efficacy of the proposed methods.
