Finite-Time Stabilization and Optimal Feedback Control for Nonlinear Discrete-Time Systems

Finite time stability involves dynamical systems whose trajectories converge to an equilibrium state in finite time. Sufficient conditions for finite time stability have recently been developed in the literature for discrete-time dynamical systems. In this paper, we build on these results to develop a framework for addressing the problem of optimal nonlinear analysis and feedback control for finite time stability and finite time stabilization for nonlinear discrete-time controlled dynamical systems. Finite time stability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function that satisfies a difference inequality involving fractional powers and a minimum operator. This Lyapunov function can clearly be seen to be the solution to a difference equation that corresponds to a steady-state form of the Bellman equation, and hence, guaranteeing both finite time stability and optimality.