Analytic Normal Forms and Symmetries of Strict Feedforward Control Systems

This paper deals with the problem of convergence of normal forms of control systems. We identify a $n$-dimensional subclass of control systems, called \emph{special strict feedforward form}, shortly (SSFF), possessing a normal form which is a smooth (resp. analytic) counterpart of the formal normal form of Kang. We provide a constructive algorithm and illustrate by several examples including the Kapitsa pendulum and the Cart-Pole system. The second part of the paper is concerned about symmetries of single-input control systems. We show that any symmetry of a smooth system in special strict feedforward form is conjugated to a \emph{scaling translation} and any 1-parameter family of symmetries is conjugated to a family of scaling translations along the first variable. We compute explicitly those symmetries by finding the conjugating diffeomorphism. We illustrate our results by computing the symmetries of the Cart-Pole system