Series expansions for analytic systems linear in the controls

Presents a series expansion for the evolution of nonlinear systems which are analytic in the state and linear in the controls. An explicit recursive expression is obtained assuming that the input vector fields are constant. Additional simplifications take place in the analysis of systems described by second order polynomial vector fields. Sufficient conditions are derived to guarantee uniform convergence over the finite and infinite time horizon. The treatment relies only on elementary notions on analytic functions, number theory and operator norms.