In the context of Lie groups, Control Theory is primarily concerned with the study of invariant, linear, bilinear and affine control systems. For invariant systems - considering that the control functions are piecewise constant - the solutions of the system has a well known and good description. This brings us to the first objective of this work: to give an explicit description of the solution curve for the other systems under the assumption that the linear vector fields commute. These solutions are obtained as the integral curve of a convenient invariant vector field on a semidirect product...