Multi-Input Nonlinear Control Systems Linearizable via One-Fold Reduction

In this paper we study the feedback linearization of multi-input control-affine systems via a particular class of nonregular feedback transformations. We give a complete geometric characterization of systems that become static feedback linearizable after a one-fold reduction of a suitably chosen control. That problem can be seen as the dual of the linearization via invertible one-fold prolongation of a suitably chosen control (which is the simplest dynamic feedback). We discuss in detail similarities and differences of both problems. We propose necessary and sufficient conditions describing the class of systems linearizable via a one-fold reduction, and discuss when the proposed conditions can be verified (by differentiation and algebraic operations only). We provide a normal form and illustrate our results by several examples.