On nonlinear controllability of homogeneous systems linear in control

This paper considers small-time local controllability (STLC) of single- and multiple-input systems, x˙=f₀(x)+Σ_i=1^mf_iu_i where f₀(x) contains homogeneous polynomials and f₁,...,f_m are constant vector fields. For single-input systems, it is shown that even-degree homogeneity precludes STLC if the state dimension is larger than one. This, along with the obvious result that for odd-degree homogeneous systems STLC is equivalent to accessibility, provides a complete characterization of STLC for this class of systems. In the multiple-input case, transformations on the input space are applied to homogeneous systems of degree two, an example of this type of system being motion of a rigid-body in a plane. Such input transformations are related via consideration of a tensor on the tangent space to congruence transformation of a matrix to one with zeros on the diagonal. Conditions are given for successful neutralization of bad type (1,2) brackets via congruence transformations.