Controllability for a class of discrete-time Hamiltonian systems

In this paper we study the controllability for a class of discrete-time nonlinear systems which arise from a discretization of a continuous-time integrable Hamiltonian systems. We give necessary and sufficient condition for the global controllability of the discrete-time nonlinear systems. The result in this paper are inspired from ergodic theory. The basic idea is as follows : for the uncontrolled (drift) system, there exists a ergodic partition, partition of phase space into a subsets. On each of the subset the drift is ergodic i.e., system trajectory will reach any positive measure set within the subset. The aim of the control is only to steer the system from one subset of the partition to another subset.