On optimization problem in electrostatic field

The problem of optimization of charged particle beam dynamics in superposition of an axially symmetric electric and space-charge and a given external fields is considered. The space-charge field and the external field not required to be axially symmetric. The complex potential of the electric field is represented with use a Cauchy integral of a control function defined on the boundary of complex region. Analytical representation of functional variation is found and the optimality conditions are formulated.