Volterra series and geometric control theory

It is shown here that controlled differential equations which are analytic in the state and linear in the control have solutions which can be expanded in a Volterra series provided there is no finite escape time. The Volterra kernels are computed in terms of the power series expansion of the functions defining the differential equation. We also give necessary and sufficient conditions for a Volterra series to be realizable by a linear-analytic system. These conditions are particularly easy to test if the Volterra series is finite; a complete theory is worked out for this case. In the final section some applications are considered to singular control, multilinear realization theory, etc.