On the finiteness of accessibility test for nonlinear discrete-time systems
This provides a finite test for accessibility in nonlinear discrete-time systems, solving a long-standing problem for two important system classes.
The paper proves that for analytic systems on compact state spaces and rational systems, there exists a finite accessibility index r* such that accessibility can be determined by testing input sequences of length r*. Algorithms to compute r* and an upper bound are provided, enabling finite tests for accessibility.
It is shown that for two large subclasses of discrete-time nonlinear systems - analytic systems defined on a compact state space and rational systems - the minimum length $r^*$ for input sequences, called here accessibility index of the system, can be found, such that from any point $x$, system is accessible iff it is accessible for input sequences of length $r^*$. Algorithms are presented to compute $r^*$, as well as an upper bound for it, which can be computed easier, and hence provide finite tests for determination of accessibility. The algorithms also show how to construct the set of points from which the system is not accessible in any finite number of steps. Finally, some relations between generic accessibility of the system and accessibility of individual points in finite steps are given.