Local convergence of Sussmann's infinite product for analytic vector fields

In 1986, Sussmann proposed an expansion of the Chen series as an infinite product of exponentials of Hall basis elements multiplied by coefficients defined through a simple induction. He proved that this product converges locally in time for bilinear control systems associated with matrices or bounded operators. In this note, we prove that Sussmann’s product also converges locally in time for control-affine systems driven by analytic vector fields. The difficulty is that one must keep track of the structure of the involved Lie brackets, as the natural decay of the explicit coefficients is not enough to counterbalance the natural growth of the iterated Lie brackets.