Finite-time stability of discrete-time systems

We deal with some finite-time control problems for discrete-time linear systems. First we provide necessary and sufficient conditions for finite-time stability; these conditions require either the computation of the state transition matrix of the system or the solution of a certain difference Lyapunov equation (or inequality). The design problem, i.e. the problem of finding a state feedback controller, which stabilizes the closed loop system in the finite-time sense, is then addressed. The way these conditions can be solved numerically is finally considered.