Statistical control in dynamical non-linear systems

Here we discuss the application of control theory to dynamic non-linear systems. This approach can be practically used for the space focusing of classical or quantum particles, for instance, in nanolithography with the beams of cooled atoms. For the practical purposes in the place of standard speed gradient method we introduce new type of controlled systems. We demand achievement of the spatial distribution of the dynamic particles and investigate the statistical properties of the particle dynamics but not the behaviour of the single particle.