  Nonlinear control-affine systems described by ordinary differential equations
with bounded measurable input functions are considered. The problem of the
existence of periodic trajectories to these systems is formulated in the sense
of Carath\'eodory solutions. It is shown that, under the dominant linearization
assumption, the periodic boundary value problem admits a unique solution for
any admissible control. This solution can be obtained as the limit of the
proposed simple iterative scheme and Newton-type method. Under additional
technical assumptions, sufficient contraction conditions of the corresponding
generating operators are derived analytically. The proposed iterative approach
is applied for the computation of periodic solutions of a realistic chemical
reaction model with discontinuous control inputs.
