A square shape of the graph of iterates of multifunctions: a complete controllability result

Abstract:

We consider a set valued function $T:K \to {2^K}$ from a domain $K$ into itself. We look for conditions under which the graph of ${T^{(n)}}$ (the $n$th iterate of $T$) will be equal to $K \times K$ for some integer $n$. When the graph of $T$ is convex a sufficient (though not necessary) condition is that neither $T(x)$ nor ${T^{ - 1}}(x)$ are contained in the boundary of $K$ whenever $x$ is there. We show that a necessary and sufficient condition is that there are neither forward nor backward trajectories which remain in the boundary for all times. In the introduction, we remark on the significance of this problem for the study of infinite-horizon control systems.