Decomposition of three-dimensional nonlinear controlled systems (Q2703884)

The main result of the paper states a sufficient condition for a three-dimensional control system, affine w.r.t. the control, to be decomposable in the sense of being equivalent to a system of type \(\dot{x}=a_{0}(x)+a(x)v\), \(\dot{y}=b_{0}(y)+b(y)v\), \(\dot{z}=c_{0}(z)+a(z)v\). NEWLINENEWLINENEWLINEThe first part of the article presents a general procedure for the factorization of affine systems of arbitrary dimension. The procedure is based on the characterization of the completely integrable codistributions whose integrals define a factorization. This characterization is used in the second part of the article to obtain a sufficient condition for a general three-dimensional system to be decomposable. The third part of the article deals specially with those three-dimensional systems that admit the most complex canonical form, i.e., \(\dot{x}=1+zu\), \(\dot{y}=u\), \(\dot{z}=H(x,y,z)u \). This yields a two-parameter set of equivalence relations in the space of smooth functions, \(H(\cdot)\).