Halo formation and control

In this report we suggest some approaches to very intricate problem of the halo formation process. It is known that this process leads to the formation of a sufficiently compact core and a spreading cloud-halo, which surrounds the core. Our approach to this problem is based on two main objects: initial distribution functions and matrix formalism for Lie algebraic tools for time evolution of particle beam. Usage of the matrix formalism allows to investigate the influence of different forms of starting model distributions. All calculations are based on symbolic representation of necessary mapping generated by space charge forces and external control fields. This allows us to formulate the basic requirements which are necessary for halo formation, that gives us a possibility to control this process