The extension of Zubov's method to sampled data control systems described by nonlinear autonomous difference equations

Theorems are developed for difference equations analogous to those developed by Zubov for differential equations. These theorems show that it is possible to construct Liapunov functions for a certain class of sampled data control systems that will give the entire domain of asymptotic stability. They also give necessary and sufficient conditions for asymptotic stability in the large for this class of control systems. In addition to giving stability information, the Liapunov function obtained from the construction procedure gives information about the performance of the system since all points having a certain value of
\upsilon
correspond to a constant value of
\prod \min{i=n}\max{\infty} (1 + \phi(x(i)))
where φ is a positive definite quadratic form chosen to measure the performance of the system. Methods of solving for the Liapunov function are considered. An approximate technique which is particularly applicable to digital computer solution is considered in detail. Two other methods involving infinite products and infinite series are also considered.