Stabilization of Discrete Nonlinear Systems with Continuous Feedback Law
The most powerful tool is the approach of control Lyapunov function which has been employed to address various issues to nonlinear control systems, and the stabilization problem of nonlinear control systems has attracted much attention. As we know, Artstein given an important theorem  which proved that the control system exists a control Lyapunov function if and only if there is a stabilizing relaxed feedback. Of course the existence of a smooth Lyapunov function fails for a nonlinear system in general. The key result used in most of the feedback stabilizers is a well known theorem due to Sontag and Brockett . The stabilization of discrete systems is studied by means of this paper. Cai Xiushan, Tang Fengjun  also studied the same systems , but the control Lyapunov function and control law which they given are not easy to get, we will give another way to get continuous control law.
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R.W.Brockett, Asymptotic stability and feedback stabilization, in Differential Geometric Control theory, R.W. Brockett, R.S. Millman, and H.J. Sussmann, eds., Birkhauser Boston, Boston, 1983,pp: 1181-191.
Cai Xiushan , Stabilization of discrete nonlinear systems based on control Lyapunov functions. J. of Systems Engineering and Electronics,2008, 19(1):131-133.
Fengjun Tang, A note on stabilization of Discrete nonlinear systems. Mathematical Problems in Engineering,2011/30631.
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